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A229111 Expansion of the g.f. of A053723 in powers of the g.f. of A121591.

%I A229111 #64 Feb 03 2023 04:05:26
%S A229111 1,-5,35,-275,2275,-19255,163925,-1385725,11483875,-91781375,
%T A229111 688658785,-4581861025,22550427925,8852899375,-2431720493125,
%U A229111 47471706909725,-699843878180125,9141002535744625,-111232778205154375,1288777160650004375,-14372445132730778975
%N A229111 Expansion of the g.f. of A053723 in powers of the g.f. of A121591.
%C A229111 In Verrill (1999) section 2.1, t = (eta(q^5) / eta(q))^6 the g.f. of A121591 and f = eta(q^5)^5 / eta(q) the g.f. of A053723.
%C A229111 Apart from signs, this is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%H A229111 Seiichi Manyama, <a href="/A229111/b229111.txt">Table of n, a(n) for n = 1..958</a>
%H A229111 Shaun Cooper, <a href="https://arxiv.org/abs/2302.00757">Apéry-like sequences defined by four-term recurrence relations</a>, arXiv:2302.00757 [math.NT], 2023. See Table 2 p. 7.
%H A229111 Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, 2016, 2:5.
%H A229111 Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See eta p. 3.
%H A229111 L. O'Brien, <a href="https://doi.org/10.13140/RG.2.2.33912.03843">Modular forms and two new integer sequences at level 7</a>, Massey University, 2016.
%H A229111 H. Verrill, <a href="http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/1999/26.ps">Some Congruences related to modular forms</a>, Max Planck Institute, 1999.
%F A229111 n^3 * a(n+1) = -(2*n - 1)*(11*n*(n - 1) + 5) * a(n) - 125 * (n - 1)^3 * a(n-1).
%F A229111 a(n*p^k) == (p^3 + Kronecker(p, 5)) * a(n*p^(k-1)) - Kronecker(p, 5) * p^3*a(n*p^(-2)) (mod p^k). [Verrill, 1999]
%F A229111 a(n) = Sum_{k=0..n-1} (-1)^k * binomial(n-1,k)^3 * binomial(5*k-(n-1),3*(n-1)). - _Seiichi Manyama_, Sep 02 2020
%e A229111 G.f. = x - 5*x^2 + 35*x^3 - 275*x^4 + 2275*x^5 - 19255*x^6 + 163925*x^7 + ...
%t A229111 a[n_] := a[n] = Switch[n, 1, 1, 2, -5, _, (1/(n-1)^3) ((1-2(n-1)) (11(n-2) (n-1)+5) a[n-1] - 125 (n-2)^3 a[n-2])];
%t A229111 a /@ Range[21] (* _Jean-François Alcover_, Jan 13 2020 *)
%o A229111 (PARI) {a(n) = my(m = n-1); if( n<1, 0, if( n<3, [1, -5][n], -( (5*(m - 1))^3*a(n-2) + (2*m - 1)*(11*(m^2 - m) +5)*a(n-1) )/ m^3))};
%o A229111 (PARI) {a(n) = sum(k=0, n-1, (-1)^k*binomial(n-1, k)^3*binomial(5*k-(n-1), 3*(n-1)))} \\ _Seiichi Manyama_, Sep 02 2020
%Y A229111 Cf. A053723, A109064, A121591.
%Y A229111 The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
%Y A229111 For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.
%K A229111 sign
%O A229111 1,2
%A A229111 _Michael Somos_, Sep 30 2013