Lots and Lots of Ramanujan-Style Congruence miracles and Lots and Lots

                     of new Integer Sequences for the OEIS



By John Chiarelli, Rebecca Coulson, Bryan Ek, Keith Frankston, Alejandro Gin\
    ory,

 Emily Kukura, Andrew Lohr, Jinyoung Park, Ali Rostami, Xukai Yan, Mingjia Y\
    ang

                              and Anthony Zaleski



      Recall that p(n) is the number of integer-patitions of the integer n

       Leohanrd Euler famously proved that these are the coefficients of



                                infinity
                                --------'
                               '  |  |      1
                                  |  |    ------
                                  |  |         i
                                  |  |    1 - q
                                 i = 1



                   Srinivasa Ramanujan famously proved that

                1/5 p(5 n + 4), 1/7 p(7 n + 5), 1/11 p(11 n + 6)

       are all integers. These integer sequences are already in the OEIS.

         These amazing facts also led to the question, first raised by

     Freeman Dyson, to give combinatorial explanations. This was answered

                     by Frank Garvan, and George Andrews.



  In this article we will state many new such congruences for the more general

                             sequences defined by



                               infinity
                               --------'       i r
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i s
                                 |  |    (1 - q )
                                i = 1



   for r from 0 to , 3, and for s from 0 to, 3, except the trivial case r=s=0

    and also supply the first, 20, terms of many new sequences for the OEIS.



  Each theorem also raises an intriguing challenge to prove it combinatorially

by finding a way to partition the counted set into subsets with the same car\
    dinality.



                                                    n
       Let's call, c[r, s](n), the coefficient of, q , in this product.



     Note that these also have natural combinatorial meanings. For example

              The case r=1, s=1 counts so-called  overpartitions.



    We don't bother with the proofs, since for the case r=0 it was shown by

            Dennis Eichhorn and Ken  Ono, in their interesting paper

                       Congruences for partition functions

 Proceedings for a Conference in Honor of Heini Halbertstam 1, 1996, pp, 309\
    -321

available from http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/013.pdf\


that one can easily determine an N_0 such that the theorem is true if it is \
    checked

for n<=N0. It turns out that the N_0 are usually rather small, and since we \
    check it

for many values of n, and we know that there exists a rigorous proof, we don\
    't bother

                             to actually prove it.

                                                                       n
        Theorem Number , 1,  Let , c[0, 1](n), be the coefficient of, q

                            in the power series of



                                infinity
                                --------'
                               '  |  |      1
                                  |  |    ------
                                  |  |         i
                                  |  |    1 - q
                                 i = 1



               Then , 1/5 c[0, 1](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 6, 27, 98, 315, 913, 2462, 6237, 15035, 34705, 77231, 166364, 348326, 710869,

    1417900, 2769730, 5308732, 9999185, 18533944, 33845975

                                                                       n
        Theorem Number , 2,  Let , c[0, 1](n), be the coefficient of, q

                            in the power series of



                                infinity
                                --------'
                               '  |  |      1
                                  |  |    ------
                                  |  |         i
                                  |  |    1 - q
                                 i = 1



               Then , 1/7 c[0, 1](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 11, 70, 348, 1449, 5334, 17822, 55165, 160215, 441105, 1159752, 2929465,

    7142275, 16873472, 38749850, 86737678, 189672868, 405991500, 852077072,

    1756048833

                                                                       n
        Theorem Number , 3,  Let , c[0, 1](n), be the coefficient of, q

                            in the power series of



                                infinity
                                --------'
                               '  |  |      1
                                  |  |    ------
                                  |  |         i
                                  |  |    1 - q
                                 i = 1



              Then , 1/11 c[0, 1](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568,

    355805845, 1117485621, 3366123200, 9767105406, 27398618368, 74534264393,

    197147918679, 508189847045, 1279140518117

                                                                       n
        Theorem Number , 4,  Let , c[0, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'
                              '  |  |        1
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[0, 2](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 22, 233, 1694, 9802, 48168, 209341, 825214, 3003660, 10224994, 32872656,

    100562026, 294490908, 829587508, 2257107225, 5951169024, 15249323665,

    38067677242, 92775934172, 221148998684

                                                                       n
        Theorem Number , 5,  Let , c[0, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'
                              '  |  |        1
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[0, 2](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 37, 354, 2446, 13630, 65203, 277386, 1074878, 3856766, 12970915, 41265542,

    125085001, 363343034, 1016150450, 2746697220, 7199261205, 18347965526,

    45576320907, 110568622854, 262447555085

                                                                       n
        Theorem Number , 6,  Let , c[0, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'
                              '  |  |        1
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[0, 2](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4, 60, 533, 3498, 18847, 87838, 366213, 1395746, 4939496, 16417680, 51701646,

    155323286, 447615063, 1242976140, 3338375959, 8699352076, 22053616456,

    54514510492, 131658684794, 311206269624

                                                                       n
        Theorem Number , 7,  Let , c[0, 3](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'
                              '  |  |        1
                                 |  |    ---------
                                 |  |          i 3
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[0, 3](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

39, 12979, 1063818, 43747800, 1164428478, 22784788890, 353208993369,

    4553192885897, 50474682682731, 493064432482017, 4323197223030294,

    34513825331341611, 253754130701090277, 1734113922213826532,

    11099317568468441634, 66963415281817368690, 382864983800470294116,

    2084136712030192782843, 10844551068435758849121, 54126544043407632835028

                                                                       n
        Theorem Number , 8,  Let , c[0, 3](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'
                              '  |  |        1
                                 |  |    ---------
                                 |  |          i 3
                                 |  |    (1 - q )
                                i = 1



             Then , 1/17 c[0, 3](17 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2093, 2002023, 425689641, 40914135699, 2350811217042, 93372583927125,

    2797362909019602, 66940486826482710, 1331816418332209968,

    22686270458415286111, 338312576306975748120, 4494261351715951994124,

    53927564012848506066083, 591125652529115865538563,

    5974755280973095614723687, 56121638476655824544215720,

    493165769051031027345167004, 4077349614980583204126847665,

    31872892656384982312919092380, 236584178429697233813974090140

                                                                       n
        Theorem Number , 9,  Let , c[2, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/7 c[2, 1](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 80, 1728, 21776, 198928, 1456816, 9052144, 49463299, 243539760, 1099495408,

    4611205920, 18146576144, 67542995936, 239304936992, 811317542455,

    2643620028976, 8309597789920, 25275621036160, 74600683091504,

    214154123742704

                                                                       n
       Theorem Number , 10,  Let , c[2, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[2, 2](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4, 56, 452, 2720, 13504, 58328, 226344, 806080, 2674280, 8356856, 24804804,

    70395520, 192026784, 505657496, 1289970400, 3197679488, 7722003180,

    18206079416, 41987387584, 94876233760

                                                                       n
       Theorem Number , 11,  Let , c[2, 3](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 3
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[2, 3](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

53, 8667, 446606, 12723004, 247840877, 3680250614, 44442051878, 455286522085,

    4075206180547, 32564312312577, 236127994179240, 1573581865533489,

    9735837318035119, 56387384315284512, 307805113693246482,

    1592705450223292382, 7849898855714632276, 37005267230334806700,

    167452613814257577264, 729637802739559376217

                                                                       n
       Theorem Number , 12,  Let , c[3, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/7 c[3, 1](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9, 704, 18568, 289784, 3258816, 29103096, 218435272, 1429144412, 8358887944,

    44512265488, 218800366200, 1003437761520, 4330026897784, 17702454168136,

    68957012312163, 257146529591424, 921677043849752, 3186099074884088,

    10653815275522744, 34548799768974704

                                                                       n
       Theorem Number , 13,  Let , c[3, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



             Then , 1/11 c[3, 1](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

263, 40296, 2073792, 59696424, 1177964224, 17719081880, 216588121712,

    2243784078160, 20289564025136, 163638700649088, 1196574355716336,

    8035101941871684, 50058729904958800, 291754500231938048,

    1601739894668762816, 8331196971005130664, 41256083572655415408,

    195323387520262845872, 887320564690091759824, 3880069547492081381248

                                                                       n
       Theorem Number , 14,  Let , c[3, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 2](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 140, 4372, 73984, 871508, 8013329, 61316716, 406485812, 2399177940,

    12856216320, 63463329809, 291850266700, 1261446696276, 5161173486092,

    20106730572544, 74948891709490, 268414159732012, 926797528603136,

    3094673295458304, 10019224226075060

                                                                       n
       Theorem Number , 15,  Let , c[3, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 2](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10, 614, 14314, 205574, 2171328, 18439526, 132703680, 837475072, 4745973834,

    24572777830, 117770297344, 527821585088, 2230113672704, 8941555875840,

    34206187974186, 125422882748154, 442489318949194, 1507007822099110,

    4968817921255680, 15899896337578688

                                                                       n
       Theorem Number , 16,  Let , c[3, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 2](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

26, 1216, 25158, 336374, 3382362, 27692800, 193679360, 1194286326, 6638737216,

    33814083072, 159783787142, 707301012032, 2955851995572, 11735857809194,

    44501556814790, 161872354235456, 566928415732826, 1917928490490026,

    6284764197896640, 19996240549466102

        This ends this article, that took, 116.805, seconds to produce.