#Ok to post homework
#Tifany Tong, December 6th, 2020, HW #22


# Question 1

# (i) L := TreeSeqL(67, t):
# coeff(L[67], t, 12) =
# 762668035595791008261768918050546247626868131307789154916300139847357281766830345010754357481701376000000000000000

# (ii) ATreeSeq(40,4)[40] =
#  90324445150366623501655158316607196285055246080819484164096000000000

# Question 2

SeqRTchild := proc(S, N) local i, mid, s, L, ans:
mid := 1 + add(z^s/s!, s in S):
L := FunEqToSeq(mid, z, N):
seq(L[i]*i!, i = 1 .. N):
end:

# (i) 1, 2, 9, 60, 540, 6120, 83790, 1345680, 24811920, 516650400, 11992503600, 307069963200, 8598348158400, 261387760233600, 8573572885878000, 301809119163552000, 11349727401396384000, 454104511068656448000, 19261139319649202976000, 863322072620761353600000
#      OEIS: A36774

# (ii) 1, 2, 9, 64, 620, 7620, 113610, 1992480, 40194000, 916927200, 23341071600, 655922836800, 20169411662400, 673645440468000, 24285190867938000, 939899116892736000, 38870133445791648000, 1710655202853140544000, 79826043011286892320000, 3936948118406837614080000
#      OEIS: A36775

# (iii) 1, 2, 9, 64, 625, 7770, 117390, 2088520, 42771960, 991090800, 25635767850, 732235165200, 22890759391500, 777398836414200, 28501053507927000, 1121908690738836000, 47194400446765572000, 2112854517933207048000, 100302903229033765260000, 5032863920347902999360000
#      OEIS: A36776

# (iv) 1, 2, 6, 28, 205, 2070, 25410, 359800, 5798520, 105663600, 2155645800, 48657886200, 1202586485100, 32282071621800, 935338118715000, 29098128631572000, 967482670702548000, 34237946556613800000, 1284878182355343576000, 50967054449973501840000
#      Not in OEIS

# (v) 1, 0, 3, 4, 65, 300, 4200, 37240, 567000, 7459200, 130873050, 2248785000, 45820074300, 960718558800, 22551581052000, 554522292324000, 14815285064580000, 416528503687296000, 12512460971526516000, 395230114168409520000
#      Not in OEIS

# Question 3

SeqRTchildNone := proc(S, N) local i, mid, s, L, ans:
mid := add(z^s/s!, s in S):
L := FunEqToSeq(exp(z) - mid, z, N):
seq(L[i]*i!, i = 1 .. N):
end:

# (i) 1, 0, 0, 4, 5, 6, 427, 1968, 6561, 220510, 2129171, 13847736, 337904437, 5156062926, 54298310445, 1192150218496, 24147409593089, 364887230459454, 8145781717395223, 197451127561855320, 3926192043196053981, 94990207383298607710, 2610336306304411857305, 64078852103083131579744, 1726495205831435720085025, 52484675948201890653360126, 1518445334604977900122070187, 45861761590549321813006087288, 1526865236101367578721315510501, 50384530075662682488952867803630
# (ii) 1, 0, 0, 0, 5, 6, 7, 8, 2529, 11350, 36971, 104556, 10182757, 99054970, 632882265, 3303250096, 165364954369, 2689602118254, 28186612549255, 233809699635780, 7892319785766381, 179814939414630850, 2756870670036244125, 33096414795555263400, 912900463740631221025, 25500259308250835585126, 525446228008825720959987, 8542226394636207936384988, 221053236679631262456996629, 6917456518383991775275958250
# (iii) 1, 0, 0, 0, 0, 6, 7, 8, 9, 10, 13871, 60996, 195637, 546560, 1411425, 427249696, 4124980929, 26013422934, 134201794555, 614270986500, 72817031988981, 1211924018940580, 12610860108434875, 103535416865167920, 734250836621295625, 45715497965321732626, 1118451164263614198837, 17224976603614475727388, 205357624826971857095129, 2077663816106417272277280
# (iv) 1, 0, 3, 0, 60, 6, 3157, 1184, 319545, 275410, 53033321, 83703456, 13098092449, 33167158220, 4510571784825, 16803854861296, 2066927734933569, 10656494045569638, 1216790715487035979, 8295602009107807560, 895347027336888089481, 7792315471001122039000, 805563125856063419979883, 8701845909614991922610520, 870378180227805997956836425, 11404602278067389636121827626, 1112438746363268367847514778237, 17345790119923612625856930761584, 1660574626556038794681719869586525, 30316620075363168814557434112312780
# (v) 1, 2, 6, 24, 120, 726, 5299, 47776, 546921, 7889140, 136394401, 2665805088, 56527983109, 1272143206894, 30092542593975, 746083879217776, 19400233368172481, 530317219796563464, 15282749263331425861, 465396262003998042440, 14992883794257587989701, 510699344284110989954122, 18354927165740287915555401, 693811746109567482139376664, 27479365035663372539331241225, 1136209816394200462905288576876, 48889737108198130555987699902039, 2183691389126980386553090058226016, 101058267340716742644796216036535641, 4839369087144324614043894777799206870

# They are all not in the OEIS

# Question 4

                        /TreeSeqL(100,x)[100]   \
              AveAndMoms|--------------------,x,6| = 
                        \        100            /

    [36.97296376, 3.114655824, 0.01302621476, 2.988844108, 

      0.1299635991, 14.83490917]

I use this as an estimate for the limit: 

(i) Average number of leaves: 36.97296376
(ii) Standard Deviation: 3.114655824
     Third Moment: 0.01302621476
     Fourth Moment: 2.988844108
     Fifth Moment: 0.1299635991
     Sixth Moment: 14.83490917