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\newtheorem*{RC}{Rademacher's Conjecture}
\newtheorem{thm}{Theorem}
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\title{Rademacher's Infinite Partial Fraction Conjecture is False}
\author{Andrew V. Sills and Doron Zeilberger}

\begin{document}
\maketitle

``It is a capital mistake to theorise before one has data.  Insensibly one begins to twist facts to suit theories, instead of theories to suit facts."---Sherlock Holmes to Dr. Watson~\cite[p. 63]{D}.
\vskip 5mm
{\bf Important Note}: This article is accompanied by the Maple package {\tt HANS}, downloadable from \\
\texttt{http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/HANS}. \\ 

The ``front'' of this article, 
\texttt{http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/hans.html}
contains lots of supporting input and output files. 

\begin{abstract}
  In his book \emph{Topics in Analytic Number Theory}, Hans Rademacher conjectured
that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition
of the generating function for partitions of integers into at most $N$ parts exist and equal
particular values that he specifies.  Despite being open for nearly four decades, little progress
has been made toward proving or disproving the conjecture, perhaps in part due to the
difficulty in actually computing the coefficients in question.  
  In this paper, we provide a fast algorithm for calculating the Rademacher coefficients,
a large amount of data, 
direct formulas for certain collections of Rademacher coefficients, and overwhelming evidence
against the truth of the conjecture.  While the limits of the sequences of Rademacher coefficients
do not exist (the sequences oscillate and attain arbitrarily large positive and negative values),
the sequences do get very close to Rademacher's conjectured limits for certain (predictable)
indices in the sequences.
\end{abstract}

\section{Introduction}
Let $p_N(n)$ denote the number of partitions of the integer $n$ into at most $N$ parts.
The generating function of $p_N(n),$
\[ F_N(x):= \sum_{n\geq 0} p_N(n) x^n = \prod_{j=1}^N \frac{1}{1-x^j}, \]
may be decomposed into partial fractions:
\begin{equation} \label{pfd}
 \prod_{j=1}^N \frac{1}{1-x^j} = \sum_{k=1}^N \underset{\gcd(h,k)=1} {\sum_{0\leq h < k } }\sum_{l=1}^{\lfloor N/k \rfloor }
 \frac{C_{h,k,l}(N) }{ (x - e^{2\pi i h/k } )^l}.\end{equation}
We shall refer to the $C_{h,k,l}(N)$ defined by~\eqref{pfd} as the \emph{Rademacher coefficients}.

Near the end of his posthumously published masterpiece \emph{Topics in Analytic
Number Theory}~\cite[p. 302]{R}, Hans Rademacher made the following conjecture:
 
\begin{RC}
For all integers $h,k,l$ such that $0 \leq h<k$, $gcd(h,k)=1$ and $l \geq 1$,  $\lim_{N\to\infty} C_{h,k,l} (N)$ exists and equals
 \begin{equation}
R_{h,k,l}:= -2\pi \left( \frac{\pi}{12} \right)^{3/2} \frac{ e^{\pi i (s(h,k)+2hl/k)}}{k^{5/2}}
  \Delta^{l-1}_{\alpha} L_{3/2}\left( -\frac{\pi^2}{6k^2} (\alpha+1) \right) , \label{RadConjValue}
\end{equation}
evaluated at $\alpha=\frac{1}{24}$,  where
$s(h,k) = \sum_{\mu=1}^{k-1}\left( \frac{\mu}{k} - \lfloor \frac{\mu}{k} \rfloor - \frac 12\right)
\left( \frac{h\mu}{k} - \lfloor \frac{h\mu}{k} \rfloor - \frac 12\right)$ is the Dedekind sum, 
%$\omega_{h,k}$ is a complex $24k$-th root of unity~\cite[p. 269]{R}, 
$\Delta_\alpha$ is the forward difference
operator, so that 
\[ \Delta_\alpha^j f(\alpha) = \sum_{h=0}^j (-1)^h \binom{j}{h} f(\alpha+j-h), \]
and
\[ L_{3/2} (-y^2) = -\frac{1}{2\sqrt{\pi} y^2} \left( 2 \cos (2y) - \frac{\sin (2y)}{y} \right). \]
In particular, if the Rademacher conjecture would have been true then it would have followed that
\[ \lim_{N\to\infty} C_{0,1,1} (N) =  R_{0,1,1} (=- \frac{6}{25}\left( 1 + \frac{2\sqrt{3}}{5\pi} \right) = -0.292927573960\dots), \]
\[ \lim_{N\to\infty} C_{0,1,2} (N) =  R_{0,1,2} (=\frac{144}{1225}+ \frac{5616}{42875\pi} =0.1897670688440\dots) , \]
\[ \lim_{N\to\infty} C_{1,2,1} (N) = R_{1,2,1} (=- \frac{2\sqrt{6}}{25}\left( \cos\frac{5\pi}{12} - \frac{12}{5\pi}\sin\frac{5\pi}{12} \right) = 0.093882853484\dots). \]
\end{RC}

(Let us make the minor remark, for the sake of mathematical accuracy, that the floating-point approximation for the value of $R_{0,1,1}$ stated by
Rademacher~\cite[p. 302]{R} was erroneous, as were the exact values of $R_{0,1,2}$ and $R_{1,2,1}$ [for the latter he gave exactly one half
of the correct value]. These erroneous values  were quoted, without correction, by Andrews~\cite[p. 388]{A}.)

Rademacher supplied (with one error) a table of values for $C_{0,1,1}(N)$, $C_{0,1,2}(N)$, and $C_{1,2,1}(N)$ for
$N=1,2,3,4,5$, and in fact these values are not too far off from conjectured ``$N=\infty$'' case.

Rademacher began work on the book in which this conjecture appeared~\cite{R} no
later than 1944 and was still working on it at the inception of his final illness.  Thus as the final version was edited and published by Rademacher's students Emil Grosswald, Joseph Lehner, and Morris Newman,
after Rademacher's death in 1969, we will never know whether Rademacher came to doubt the truth of the conjecture after
he had written it down.  However, George Andrews reports that Rademacher discussed 
the conjecture in a course he taught at the University of Pennsylvania during the 1961--1962
academic year.

In~\cite{M}, Augustine Munagi considered a different type of partial fraction decomposition
called $q$-partial fractions, and proved a special case of the analog of the Rademacher
conjecture, relative to the $q$-partial fraction decomposition. 

We should also note that the first to cast doubts on the Rademacher conjecture were Jane Friedman and
Leon Ehrenpreis. Ehrenpreis~\cite[p. 317]{E} stated,  
%``If one attempts to carry out the usual type of
%partial fraction decomposition of the partition function term-by-term, it is difficult to compute the
%coefficients.  
``My student, Jane Friedman, spent a great deal of time trying to apply computer algorithm
methods to compare the coefficients with those of Rademacher\dots Unfortunately, the
computer study proved inconclusive.'' 

In this
article, we present overwhelming evidence against this conjecture, taken literally, but we will present ample
evidence for a modified conjecture.    Additionally, we will present a fast algorithm for 
generating the Rademacher coefficients, and formulas for a selection of particular 
Rademacher coefficients.

\section{Empirical evidence against the Rademacher conjecture}
\subsection{The \emph{actual} behavior of the sequence $C_{0,1,1}(N)$}

The values of $C_{0,1,1}(N)$, for $1\leq N \leq 700$ are provided, in both exact
rational and floating point approximation form, at
\\
\texttt{http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/oHANS1}\\

Figures 1 and 2 are graphical summaries of $C_{0,1,1}(N)$.
%Here are the approximate values of $C_{0,1,1}(N)$ for $N=1,2,\dots,300$:
%\begin{verbatim}
%-1., -0.2500000000, -0.2361111111, -0.2361111111, -0.2410995370, -0.2483304398,
%   -0.2566047939, -0.2652668959, -0.2738962471, -0.2821915981, -0.2899214161,
%   -0.2969011397, -0.3029824976, -0.3080486870, -0.3120125144, -0.3148160158,
%   -0.3164307261, -0.3168581002, -0.3161297599, -0.3143073433, -0.3114818012,
%   -0.3077720281, -0.3033227577, -0.2983016835, -0.2928957998, -0.2873069878,
 %  -0.2817469065, -0.2764312745, -0.2715736626, -0.2673789445, -0.2640365764,
%   -0.2617139016, -0.2605496910, -0.2606481443, -0.2620735798, -0.2648460412,
%   -0.2689380368, -0.2742726128, -0.2807229297, -0.2881134797, -0.2962230389,
%   -0.3047893943, -0.3135158330, -0.3220793139, -0.3301401806, -0.3373532060,
%   -0.3433796927, -0.3479002934, -0.3506281567, -0.3513219589, -0.3497983445,
 %  -0.3459432771, -0.3397217949, -0.3311856770, -0.3204785572, -0.3078380724,
  % -0.2935947031, -0.2781670532, -0.2620534253, -0.2458196715, -0.2300834357,
  % -0.2154950505, -0.2027155028, -0.1923920297, -0.1851320536, -0.1814762907,
  % -0.1818719855, -0.1866473053, -0.1959879875, -0.2099173533, -0.2282807762,
  % -0.2507356340, -0.2767476573, -0.3055944315, -0.3363766061, -0.3680371175,
  % -0.3993884510, -0.4291476533, -0.4559784752, -0.4785396777, -0.4955381981,
  % -0.5057855456, -0.5082555026, -0.5021409611, -0.4869075360, -0.4623414838,
  % -0.4285894287, -0.3861874704, -0.3360774211, -0.2796082062, -0.2185208537,
  % -0.1549159998, -0.09120342809, -0.03003384264, 0.02578618604, 0.07339437807,
  % 0.1099854828, 0.1329332966, 0.1399179263, 0.1290536201, 0.09901196342,
  % 0.04913487021, -0.02046838146, -0.1088457811, -0.2141543228, -0.3336437550,
  % -0.4636764597, -0.5997865313, -0.7367798352, -0.8688753194, -0.9898861679,
  % -1.093437573, -1.173216021, -1.223243100, -1.238165066, -1.213547722,
  % -1.146164839, -1.034267255, -0.8778191962, -0.6786881978, -0.4407754663,
  % -0.1700745026, 0.1253525123, 0.4354888056, 0.7485729229, 1.051406370,
  % 1.329756984, 1.568851754, 1.753948314, 1.870969758, 1.907182998,
  % 1.851896756, 1.697151715, 1.438372519, 1.074949423, 0.6107167168,
  % 0.05429558449, -0.5807288755, -1.275823185, -2.007735196, -2.748952948,
  % -3.468399620, -4.132362211, -4.705641343, -5.152898598, -5.440166609,
  % -5.536476038, -5.415543178, -5.057452686, -4.450262444, -3.591452261,
  % -2.489135606, -1.162954149, 0.3554209099, 2.022248908, 3.782197526,
  % 5.569379102, 7.308966976, 8.919390911, 10.31508556, 11.40973887,
  % 12.11995947, 12.36925404, 12.09217956, 11.23851097, 9.777245540,
  % 7.700250311, 5.025351315, 1.798663217, -1.904033469, -5.977042356,
  % -10.28577535, -14.66916045, -18.94349589, -22.90775776, -26.35030638,
  % -29.05686865, -30.81960342, -31.44698622, -30.77418146, -28.67350904,
  % -25.06455928, -19.92347164, -13.29086909, -5.277936872, 3.929847828,
  % 14.07178633, 24.81421369, 35.75623110, 46.43909210, 56.35928117,
  % 64.98516624, 71.77693593, 76.20935250, 77.79667005, 76.11889247,
  % 70.84838391, 61.77570729, 48.83346001, 32.11681125, 11.89942875,
   %-11.35647811, -36.99717219, -64.18246914, -91.89966584, -118.9868068,
  % -144.1654288, -166.0825264, -183.3610397, -194.6577089, -198.7266644,
  % -194.4866703, -181.0895127, -157.9866611, -124.9910454, -82.33060818,
  % -30.69023753, 28.76123477, 94.36203911, 163.9691435, 234.9926874,
  % 304.4545217, 369.0713099, 425.3616182, 469.7752894, 498.8421968,
  % 509.3362499, 498.4493245, 463.9686793, 404.4504443, 319.3810057,
  % 209.3176002, 75.99925493, -77.58061224, -247.1480216, -427.1757573,
  % -610.9697867, -790.8184690, -958.2059363, -1104.088366, -1219.228913,
  % -1294.583919, -1321.729828, -1293.317032, -1203.533964, -1048.562131,
  % -827.0007167, -540.2379967, -192.7462527, 207.7227341, 650.0630467,
  % 1119.879686, 1599.708353, 2069.400076, 2506.674746, 2887.840680,
  % 3188.669548, 3385.407725, 3455.896649, 3380.766336, 3144.658365,
  % 2737.427674, 2155.266899, 1401.693204, 488.3358512, -564.5363131,
   %-1727.804587, -2963.657762, -4226.157644, -5462.240380, -6613.165381,
  % -7616.405236, -8407.949469, -8924.973071, -9108.798049, -8908.053911,
  % -8281.921848, -7203.328711, -5661.941659, -3666.803884, -1248.447020,
  % 1539.682196, 4620.633996, 7894.341889, 11239.09288, 14514.15716,
  % 17563.61103, 20221.33767, 22317.13617, 23683.81017, 24165.04726,
  % 23623.84008, 21951.14326, 19074.40991, 14965.61003, 9648.304228,
  % 3203.332803, -4227.315939, -12438.86960, -21164.73363
% \end{verbatim}
 
 \begin{center}
  \begin{figure}[h] \caption{Graph of $C_{0,1,1}(N)$ for $N$ from $1$ to
 $100$, together with the line $y=-0.2929276$ }
 \includegraphics[scale=0.75]{C011-100graphic}
\end{figure}
 
 \begin{figure}[h] \caption{Graph of $C_{0,1,1}(N)$ for $N$ from $1$ to
 $200$. }
 \includegraphics[scale=0.75]{C011-200graphic.pdf}
\end{figure}
\end{center}

Notice that $C_{0,1,1}(25)$ differs from Rademacher's conjectured value of
$C_{0,1,1}(\infty)$ by less than $0.000032$, but then things go down hill (for
the conjecture) from there, and get particularly bad after about $n=150$.
Numerical evidence points to the sequence $C_{0,1,1}(N)$ oscillating
and attaining arbitrarily large positive and negative values.
The same thing is true for other sequences $C_{h,k,l}(N)$; see the output at the webpage of this article

\texttt{http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/hans.html} \\


By examining the graphs in Figures 1 and 2 and the associated numerical
data, it seems reasonable to
state the following alternative conjecture:
\begin{conj}
 $C_{0,1,1}(N)$ is an oscillating function of $N$ of ``period" about $32$, whose absolute value increases
 without bound as $N$ increases.
\end{conj}

By ``period'' $32$ we mean that the peaks and valleys,  eventually, recur at a period of $32$. We also
noticed, numerically, that the elevations and depths of successive peaks and valleys roughly grows
exponentially with a factor around $8$.
\vskip 3mm

\subsection{How to compute the sequence $C_{0,1,1}(N)$ fast} 

If you use the {\bf definition} of $C_{0,1,1}(N)$, or Andrews's formula~\cite[p. 388, Theorem 1]{A}, you can't go very far, even with Maple.
Rademacher calculated $C_{0,1,1}(N)$ for $N=1,2,3,4,5$, presumably by hand, and made
an error in the $N=5$ case.
Andrews~\cite[p. 388]{A}, who had access to a computer algebra system in 2003, 
corrected Rademacher's error at $N=5$ and extended the
list to $N=6,7,8$.

We need to be
more clever. A fast recurrence can be derived as follows.
Since
$$
\prod_{j=1}^N \frac{1}{1-x^j}=\sum_{l=1}^{N} \frac{C_{0,1,l}(N)}{(x-1)^l} + \dots  \quad ,
$$
we can multiply both sides by $(x-1)^N$ and get
$$
(x-1)^N\prod_{j=1}^N \frac{1}{1-x^j}=\sum_{r=0}^{N-1} D_r(N){(x-1)^r} + \dots  \quad .
$$
Once we know $D_r(N)$, we can find $C_{0,1,l}(N)$, since they equal $D_{N-l}(N)$. It remains to
find a fast recurrence for $D_r(N)$. \\

By definition, we have:
$$
\frac{1-x^N}{x-1} \left ( \sum_{r=0}^{\infty} D_r(N){(x-1)^r} \right )=
 \sum_{r=0}^{\infty} D_r(N-1){(x-1)^r} \quad.
$$
Letting $z=x-1$, this is:
$$
\frac{1-(z+1)^N}{z} \left ( \sum_{r=0}^{\infty} D_r(N){z^r} \right )=
 \sum_{r=0}^{\infty} D_r(N-1){z^r} \quad.
$$
By the binomial theorem,
$$
- \left ( \sum_{a=0}^{N-1} \binom{N}{a+1} z^a \right ) \left ( \sum_{r=0}^{\infty} D_r(N){z^r} \right )=
 \sum_{r=0}^{\infty} D_r(N-1){z^r} \quad.
$$
Equating coefficients of $z^r$ we get:
$$
ND_r(N)+\sum_{a=1}^{r} \binom{N}{a+1} D_{r-a}(N)= -D_r(N-1) \quad .
$$
And finally:
$$
D_r(N)=-\frac{D_r(N-1)}{N}-\sum_{a=1}^{r} \frac{1}{N}\binom{N}{a+1} D_{r-a}(N) \quad .
$$
This is implemented in procedure {\tt C01(l,N)} of {\tt HANS}.

The same argument leads to efficient recurrences for $C_{h,k,l}(N)$, except that now we have
to distingish between the case when $N$ is divisible by $k$ and when it is not, yielding
two different recurences. This is implemented in procedure {\tt ChklN(h,k,l,N)} of {\tt HANS}. \\


\subsection{$C_{1,2,1}(N)$}

The values of $C_{1,2,1}(N)$ for $1\leq N\leq 700$, in both exact rational form and
approximate floating point form are provided at

\texttt{http://www.math.rutgers.edu/~zeilberg/tokhniot/oHANS3}

Graphical summaries are provided in Figures 3 and 4.

 \begin{figure}[h] \caption{Graph of $C_{1,2,1}(N)$ for $N$ from $1$ to
 $150$, together with the line $y=0.09388285$ }
 \includegraphics[scale=0.75]{C121-150graphic}
\end{figure}

 \begin{figure}[h!] \caption{Graph of $C_{1,2,1}(N)$ for $N$ from $1$ to
 $300$, together with the line $y=0.09388285$ }
 \includegraphics[scale=0.75]{C121-300graphic}
\end{figure}


\section{``Top down" formulas for the Rademacher Coefficients}
\subsection{$C_{0,1,l}(N)$}

As Rademacher already pointed out, it seems hopeless to get a closed-form formula for
$C_{h,k,l}(N)$ for $l=1,2, \dots$, {\it but} if you work your way up from the ``top''
one can conjecutre, and then {\it rigorously} prove explicit formuals for
$C_{h,k,N-r}(N)$, that alas, get increasingly more complicated as $r$ gets larger.


\begin{conj} \label{C01topdown}
 \[ C_{0,1,N-r}(N)  =  \frac{(-1)^{N+r}}{4^r N! r!} P_{0,1,N-r} (N), \]
 where $P_{0,1,N-r}(N)$ is a convex, alternating, monic polynomial of degree $2r$ whose
 only real roots are $0$ and $1$.
\end{conj}

\begin{thm}  \label{P01topdown}Explicit formulas for the $P_{0,1,N-r}(N)$ of Conjecture~\ref{C01topdown} may
be given for any specific $r$,
in particular, we have
 \begin{equation} \label{P01N} P_{0,1,N}(N) = 1 \end{equation}
\begin{equation} \label{P01N-1}  P_{0,1,N-1}(N)= N^2 - N \end{equation}
\begin{equation} \label{P01N-2} P_{0,1,N-2}(N)= N^4-\frac{22 N^3}{9}+\frac{13 N^2}{3}-\frac{26 N}{9} \end{equation}
% \begin{equation} \label{P01N-3} P_{0,1,N-3}(N)=N^6-\frac{13 N^5}{3}+\frac{43 N^4}{3}-25 N^3+\frac{98 N^2}{3}-\frac{56 N}{3} \end{equation}
%   \[ P_{0,1,N-4}(N) = N^8-\frac{20 N^7}{3}+\frac{862 N^6}{27}-\frac{21104 N^5}{225}+\frac{29039
%   N^4}{135}-\frac{14548 N^3}{45}+\frac{9892 N^2}{27}-\frac{42896 N}{225}\]
 %  \begin{multline*}
 %  P_{0,1,N-5}(N) = 
 %  N^{10}-\frac{85 N^9}{9}+\frac{1610 N^8}{27}-\frac{33862 N^7}{135}+\frac{36869
 %  N^6}{45}-\frac{269371 N^5}{135}+\frac{34712 N^4}{9} \\-\frac{716548
 %  N^3}{135}+\frac{742928 N^2}{135}-\frac{120448 N}{45}
 %  \end{multline*}
%\begin{multline*}
%P_{0,1,N-6}(N) = 
%  N^{12}-\frac{38 N^{11}}{3}+\frac{905 N^{10}}{9}-\frac{675574 N^9}{1215}+\frac{321073
 %  N^8}{135}-\frac{158916134 N^7}{19845}\\+\frac{4121695 N^6}{189}-\frac{134142298
  % N^5}{2835}+\frac{11129492 N^4}{135}-\frac{908622824 N^3}{8505}\\+\frac{2805920
  % N^2}{27}-\frac{317019904 N}{6615}
%\end{multline*}
%\begin{multline*}
%P_{0,1,N-7}(N) = 
%N^{14}-\frac{49 N^{13}}{3}+\frac{4277 N^{12}}{27}-\frac{1328341
 %  N^{11}}{1215}+\frac{7092211 N^{10}}{1215}-\frac{4750241 N^9}{189}+\frac{251909783
  % N^8}{2835}\\-\frac{739666699 N^7}{2835}+\frac{257315158 N^6}{405}-\frac{1545607364
  % N^5}{1215}+\frac{2507097128 N^4}{1215}-\frac{7249838912 N^3}{2835}\\+\frac{447420800
  % N^2}{189}-\frac{328999936 N}{315}
%\end{multline*}
 \end{thm}
 
 \begin{rmk}
 Readers desiring formulas for $C_{0,1,N-r}(N)$ for $r>3$ are directed to the
\texttt{ChkFormula} procedure in the
 \texttt{HANS}
Maple package.
 \end{rmk}

\begin{rmk}
From the above (and additional data not reproduced here but available at the website), we may deduce that
\begin{multline*}
   P_{0,1,N-r}(N) = N^{2r} - \frac{2r^2+7r}{9} N^{2r-1} + \frac{ 2r^4 + 6r^3 + \frac{287}{2}r^2 - \frac{303}{2}r }{9^2} N^{2r-2} + \mbox{ lower order terms.} 
\end{multline*}
\end{rmk}

 
 \begin{proof}[Proof of Theorem~\ref{P01topdown}]
 Define $G_N := (x-1)^N F_N(x).$  Then $G_N$ has a Taylor series expansion about $x=1$, whose first $N$
 coefficients are the Rademacher coefficients:
 \[ G_N = \sum_{j=0}^{N-1} C_{0,1,N-j}(N)  (x-1)^j + O( (x-1)^N ). \]
 Clearly, \begin{equation} \label{Grec} (1-x^N) G_N = (x-1) G_{N-1}. \end{equation}
 Expanding $(1-x^N)$ on the left hand side of~\eqref{Grec} as a Taylor polynomial about $x=1$, we have
 \begin{multline} \label{C011rec}
  \left( -\sum_{j=1}^N \binom{N}{j} (x-1)^j \right) \left( \sum_{j=0}^{N-1} C_{h,k,N-j}(N)  (x-1)^j + \mbox{ higher degree terms } 
  \right)
   \\ = \sum_{j=1}^{N-1} C_{0,1,N-j}(N-1) (x-1)^j + \mbox{higher degree terms}
 \end{multline}
 
Comparing the coefficients of $(x-1)^1$ on both sides of~\eqref{C011rec}, we find
\[ -N C_{0,1,N}(N) = C_{0,1,N-1}(N-1). \]  Solving the recurrence with the initial condition $C_{0,1,1}(1) = -1$,
yields 
\begin{equation} \label{C01NN} C_{0,1,N} (N) = \frac{(-1)^{N}}{N!}, \end{equation} which is~\eqref{P01N}.
 
Comparing the coefficients of $(x-1)^2$ on both sides of~\eqref{C011rec}, we find, taking into account~\eqref{C01NN},
\begin{equation}  -N C_{0,1,N-1}(N) - \binom N2 \frac{(-1)^N}{N!} =  C_{0,1,N-2}(N-1)  \end{equation}
with initial condition $C_{0,1,1}(2) = -\frac{1}{4}$ yields
\begin{equation} \label{C01N-1N} C_{0,1,N-1}(N) = \frac{(-1)^{N+1}}{4(N-2)!} , \end{equation}
which is~\eqref{P01N-1}.

Comparing the coefficients of $(x-1)^3$ on both sides of~\eqref{C011rec}, we find, taking into account~\eqref{C01NN}
and~\eqref{C01N-1N},
\begin{equation}  -N C_{0,1,N-2}(N) - \binom N2 \frac{(-1)^{N+1}}{4(N-2)!} 
-\binom N3 \frac{(-1)^N}{N!} =  C_{0,1,N-3}(N-1)  \end{equation}
with initial condition $C_{0,1,1}(3) = -\frac{17}{72}$ yields
\begin{equation} \label{C01N-2N} C_{0,1,N-2}(N) = \frac{(-1)^{N}(9N^2 - 13N + 26)}{288(N-2)!} , \end{equation}
which is~\eqref{P01N-2}.
Results for larger $r$ follow analogously.
\end{proof} 


\subsection{$C_{1,2,l}(N)$}
Let us now define $\bar{G}_N$, analogous to $G_N$.
Let $$\bar{G}_N := (x+1)^{\lfloor N/2 \rfloor} F_N(x).$$  Then $\bar{G}_N$ has a Taylor series expansion about $x=-1$,
whose first $\lfloor \frac N2 \rfloor$ coefficients are the Rademacher coefficients:
\[ \bar{G}_N = \sum_{r=0}^{\lfloor N/2 \rfloor - 1} C_{1,2,\lfloor N/2 \rfloor - r}(N) (x+1)^r + O( (x+1)^{\lfloor N/2 \rfloor} ) .\]
We now abandon the use of the floor function.
Notice that 
\begin{equation} 
   (1-x^{2n-1})(1-x^{2n}) \bar{G}_{2n} = (x+1)\bar{G}_{2n-2}.
\end{equation} %and
%\begin{equation}
%  (1-x^{2j})\bar{G}_{2j-1} = (x+1) \bar{G}_{2j}.
%\end{equation}
Thus, by expanding the two left most factors on the left side as a Taylor series about $x=-1$,
\begin{multline} \label{C12even}
  \left\{ \sum_{r=1}^{4n-1} (-1)^{r+1} \left[ \binom{4n-1}{r} + \binom{2n}{r} - \binom{2n-1}{r} \right] (x+1)^r \right\} \\
  \times \left( \sum_{r=0}^{n-1} C_{1,2,n-r}(2n) (x+1)^r + \mbox{ higher degree terms }\right) \\
 = \sum_{r=1}^{n-1} C_{1,2,n-r} (2n-2) (x+1)^r + \mbox{ higher degree terms } 
\end{multline}
%and
%\begin{multline} \label{C12oddtoeven}
%  \left( \sum_{r=1}^{2j} (-1)^{r+1} \binom{2j}{r} (x+1)^r \right) 
%  \left( \sum_{r=0}^{j-1} C_{1,2,j-r}(2j) (x+1)^r + \mbox{ higher order terms }\right) \\
% = \sum_{r=1}^{j-1} C_{1,2,j-r} (2j-1) (x+1)^r + \mbox{ higher order terms. }
%\end{multline}

By comparing coefficients of $(x+1)^r$ in both sides of ~\eqref{C12even} 
%\eqref{C12eventoodd} and
%\eqref{C12oddtoeven}, 
and solving the recurrences, we obtain formulas
for $C_{1,2,n-r}(2n)$
%and $C_{1,2,j-r}(2j)$ 
analogous to those for $C_{0,1,N-r}(N)$.

% Comparing coefficients of $(x+1)$ on both sides of~\eqref{C12even}, we find
% \[ 4nC_{1,2,n}(2n) = C_{1,2,n-1}(2n-2), \]
% which, together with initial condition $C_{1,2,1}(2) = \frac 14$, yields
 \begin{equation} \label{C12n2n}
    C_{1,2,n}(2n) = \frac{1}{2^{2n} n!}.
 \end{equation}

%Comparing coefficients of $(x+1)^2$ on both sides of~\eqref{C12even}, and taking into account~\eqref{C12n2n}, we find
% \[ 4nC_{1,2,n-1}(2n) - \frac{2n-1}{4^{n-1} (n-1)!} = C_{1,2,n-2}(2n-2), \]
% which, together with initial condition $C_{1,2,1}(4) = \frac 18$, yields
 \begin{equation} \label{C12n-12n}
    C_{1,2,n-1}(2n) = \frac{n}{2^{2n} (n-1)!}.
 \end{equation}

%Comparing coefficients of $(x+1)^3$ on both sides of~\eqref{C12even}, and taking into account
%\eqref{C12n2n} and~\eqref{C12n-12n}, we find
% \[ 4nC_{1,2,n-2}(2n) - \frac{4n^2(2n-1)}{4^{n} (n-1)!} + \frac{n(2n-1)(16n-13)}{3\cdot 4^n n!} = C_{1,2,n-3}(2n-2), \]
% which, together with initial condition $C_{1,2,1}(6) = \frac {461}{4608}$, yields
 \begin{equation} \label{C12n-22n}
    C_{1,2,n-2}(2n) = \frac{18n^3 - 8n^2 + 15n + 2}{3^2\cdot 2^{2n+2} (n-1)!}.
 \end{equation}

%This process can be continued indefinitely with higher powers of $(x+1)$, e.g.
%\begin{equation}C_{1,2,n-3}(2n) = \frac{(n^2+1)(6n^3-8n^2+9n+2)}{3^2\cdot 2^{2n+2} (n-1)!},\end%{equation}
%\begin{equation} C_{1,2,n-4}(2n) = \frac{ 2700n^7 - 7200n^6 + 15100n^5 - 17448n^4+21205n^3 %-8880n^2+
%13350n+3423}{25\cdot 3^4\cdot 2^{2n+5} (n-1)!}. \end{equation}

Of course, the observation 
$ (1-x^{2n})(1-x^{2n+1}) \bar{G}_{2n+1} = (x+1)\bar{G}_{2n-1} $
leads to analogous formulas for the $C_{1,2,n-r}(2n+1)$, e.g.,
\begin{equation}
 C_{1,2,n}(2n+1)= \frac{1}{2^{2n+1} \ n!},
\end{equation}
\begin{equation}
 C_{1,2,n-1}(2n+1)= \frac{2n^2+2n+1}{2^{2n+2}\ n!},
\end{equation}
\begin{equation}
C_{1,2,n-2}(2n+1) = \frac{18n^5+46n^4 + 61n^3 + 53n^2 + 29n + 9}{9\cdot 2^{2n+3}\  (n+1)!},
\end{equation}
%\begin{equation}
%C_{1,2,n-3}(2n+1) = \frac{(2n^2+2n+3)(6n^4+4n^3+3n^2+2n+3)}{9\cdot 2^{2n+4} n!}.
%\end{equation}

Clearly, the same idea can be used to find formulas for $C_{h,k,n-j}(kn+r)$ for any $h$, $k$, $j$, $r$.
This has been implemented in the procedure \texttt{ChkFormula} in the \texttt{HANS} Maple package.
For those desiring automatically generated papers, containing both formulas of this type and their
proofs, please use the \texttt{HansTopDownAutoPaper} procedure in the \texttt{HANS} Maple package.


\section{Close Encounters of the Rademacher Kind}
While it appears that $\lim_{N\to\infty} C_{h,k,l}(N)$ does not exist for any $(h,k,l)$,  
we can nonetheless define  $B_{h,k,l}$ to be the $N$ which comes closest to 
Rademacher's conjectured value of $C_{h,k,l}(\infty)$.  This is implemented in
the \texttt{CloseEncounters} procedure in \texttt{HANS}.

\begin{center}
\begin{tabular}{| c | c | r | r | }
$l$ & $B_{0,1,l}$ & absolute difference & absolute ratio  \\
\hline
 $1$ &  $25$ & $0.0003177 $ & $0.99989 $\\
 $2$ &  $47$ & $0.0001434 $ & $0.99924 $\\
 $3$ &  $71$ & $0.0000828 $ & $0.99991 $\\
 $4$ &  $149$ & $0.0000009$ & $1.00001$ \\
 %$5$ &  $109$ &  $   $ & $   $\\
\end{tabular}
\end{center}

Notice that the first few values of $B_{0,1,l}$ are close to $24l$.  This motivates us to 
consider comparing $C_{0,1, l}(24l)$ to $R_{0,1,l}$,
Rademacher's conjectured value $R_{0,1,l}$ of $C_{0,1,l}(\infty)$.

\begin{center}
\begin{tabular}{| c | l | l | }
$l$ & $\vert C_{0,1,l}(24l) - R_{0,1,l}\vert$ &  $\vert C_{0,1,l}(24l) / R_{0,1,l} \vert $ \\
\hline
 $1$ & $0.0053741095$ & $ 1.018346206$ \\
 $2$ & $0.0015044594$ & $ 1.007927400$ \\
 $3$ & $0.00033240887$ & $ 0.996241370$ \\
 $4$ & $0.00004427030$ & $ 1.001376635$ \\
 $5$ & $0.000011288321$ & 0.9988220859 \\
 $6$ & $0.000001686611$ &1.0006971253\\
 $7$ & $0.0000001275687$ & 0.9997575030\\
 $8$ & $0.0000000110523$ &    1.0000986383 \\
 $9$ & $0.00000000239242$ & 0.9999562770 \\
 $10$ & $0.000000005333208$ & 1.0000141594\\
 $11$ & $0.0000000187490584$ & 0.9999947242\\
 $12$ & $0.0000000393434274$ & 1.0000017401 \\
 \end{tabular}
 \end{center}
 
 Thus we have some evidence that even though the $N$ for which $C_{0,1,l}(N)$ is closest
 to $R_{0,1,l}$ is not $N=24l$, $C_{0,1,l}(24l)$ seems to provide a good approximation to
 $R_{0,1,l}$.


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\end{document}