%I #28 Sep 08 2022 08:46:18
%S 1,2,22,336,6006,117348,2428272,52303680,1160427510,26337699740,
%T 608642155660,14272471122560,338764038330480,8123136091556640,
%U 196484811079765440,4788469475873867520,117465323079289162230,2898183118626011393100
%N Expansion of g.f. of A002652 in powers of the g.f. of A279618.
%C G.f. is the square root of the g.f. for A183204.
%C This sequence is c_n in Theorem 6.1 in O'Brien's thesis.
%C Also see Conjecture 5.4 in Chan, Cooper and Sica's paper.
%D L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.
%H G. C. Greubel, <a href="/A279619/b279619.txt">Table of n, a(n) for n = 1..500</a>
%H H. H. Chan, S. Cooper, F. Sica, <a href="http://www.francescosica.org/Francesco_Sica/Publications_files/chancoopersica.pdf">Congruences satisfied by Apéry-like numbers</a>, International Journal of Number Theory, 2010, 6(01), 89-97. Conjecture 5.4.
%H Lynette O'Brien, <a href="https://www.researchgate.net/profile/Lynette_Obrien">Modular forms and two new integer sequences at level 7</a>
%H Lynette O'Brien, <a href="/A279619/a279619.pdf">Modular forms and two new integer sequences at level 7</a>
%F (n+1)^2*a_7(n+1) = (26*n^2+13*n+2)*a_7(n) + 3*(3*n-1)*(3*n-2)*a_7(n-1), a(0)=1, a(-1)=0.
%F Conjecture: For any positive integer n and any prime p with p equiv. 0,1,2 or 4 modulo 7, a(n) equiv. a(n)=a(n_0)a(n_1)...a(n_r) modulo p, where n=n_0+n_1p+...n_rp^r is the base p representation of n.
%F Conjecture: a(n)~ C n^(-3/2) 27^n where C=0.0955223052681267146513079107870296256727946666510071798669948234917659...
%e G.f. = 1 + 2*x + 22*x^2 + 336*x^3 + 6006*x^4 + ....
%t RecurrenceTable[{a[n+1] == ((26*n^2+13*n+2)*a[n] + 3*(3*n-1)*(3*n-2)*a[n-1])/ (n + 1)^2, a[-1] == 0, a[0] == 1}, a, {n, 0, 50}] (* _G. C. Greubel_, Jul 04 2018 *)
%t CoefficientList[Series[Sqrt[7]*(1/(25 - 80*x + 24*Sqrt[1 - 27*x]*Sqrt[1+x]))^(1/4) * Hypergeometric2F1[1/12, 5/12, 1, 13824*x^7/(1 - 21*x + 8*x^2 + Sqrt[1 - 27*x] * (1 - 8*x)*Sqrt[1+x])^3], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jul 04 2018 *)
%o (Magma) I:=[2, 22]; [1] cat [n le 2 select I[n] else ((26*n^2-39*n+15)* Self(n-1) + 3*(3*n-4)*(3*n-5)*Self(n-2))/n^2 : n in [1..50]] // _G. C. Greubel_, Jul 04 2018
%Y Cf. A183204, A279613, A279618.
%Y 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.)
%K nonn
%O 1,2
%A _Lynette O'Brien_, Dec 15 2016