This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A279619 #28 Sep 08 2022 08:46:18
%S A279619 1,2,22,336,6006,117348,2428272,52303680,1160427510,26337699740,
%T A279619 608642155660,14272471122560,338764038330480,8123136091556640,
%U A279619 196484811079765440,4788469475873867520,117465323079289162230,2898183118626011393100
%N A279619 Expansion of g.f. of A002652 in powers of the g.f. of A279618.
%C A279619 G.f. is the square root of the g.f. for A183204.
%C A279619 This sequence is c_n in Theorem 6.1 in O'Brien's thesis.
%C A279619 Also see Conjecture 5.4 in Chan, Cooper and Sica's paper.
%D A279619 L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.
%H A279619 G. C. Greubel, <a href="/A279619/b279619.txt">Table of n, a(n) for n = 1..500</a>
%H A279619 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 A279619 Lynette O'Brien, <a href="https://www.researchgate.net/profile/Lynette_Obrien">Modular forms and two new integer sequences at level 7</a>
%H A279619 Lynette O'Brien, <a href="/A279619/a279619.pdf">Modular forms and two new integer sequences at level 7</a>
%F A279619 (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 A279619 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 A279619 Conjecture: a(n)~ C n^(-3/2) 27^n where C=0.0955223052681267146513079107870296256727946666510071798669948234917659...
%e A279619 G.f. = 1 + 2*x + 22*x^2 + 336*x^3 + 6006*x^4 + ....
%t A279619 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 A279619 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 A279619 (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 A279619 Cf. A183204, A279613, A279618.
%Y A279619 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 A279619 nonn
%O A279619 1,2
%A A279619 _Lynette O'Brien_, Dec 15 2016