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 A279619 Expansion of g.f. of A002652 in powers of the g.f. of A279618. 36
 1, 2, 22, 336, 6006, 117348, 2428272, 52303680, 1160427510, 26337699740, 608642155660, 14272471122560, 338764038330480, 8123136091556640, 196484811079765440, 4788469475873867520, 117465323079289162230, 2898183118626011393100 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS G.f. is the square root of the g.f. for A183204. This sequence is c_n in Theorem 6.1 in O'Brien's thesis. Also see Conjecture 5.4 in Chan, Cooper and Sica's paper. REFERENCES L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016. LINKS G. C. Greubel, Table of n, a(n) for n = 1..500 H. H. Chan, S. Cooper, F. Sica, Congruences satisfied by Apéry-like numbers, International Journal of Number Theory, 2010, 6(01), 89-97. Conjecture 5.4. Lynette O'Brien, Modular forms and two new integer sequences at level 7 Lynette O'Brien, Modular forms and two new integer sequences at level 7 FORMULA (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. 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. Conjecture: a(n)~ C n^(-3/2) 27^n where C=0.0955223052681267146513079107870296256727946666510071798669948234917659... EXAMPLE G.f. = 1 + 2*x + 22*x^2 + 336*x^3 + 6006*x^4 + .... MATHEMATICA 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 == 1}, a, {n, 0, 50}] (* G. C. Greubel, Jul 04 2018 *) CoefficientList[Series[Sqrt*(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 *) PROG (MAGMA) I:=[2, 22];  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 CROSSREFS Cf. A183204, A279613, A279618. 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.) Sequence in context: A299824 A266888 A155674 * A245113 A078232 A151615 Adjacent sequences:  A279616 A279617 A279618 * A279620 A279621 A279622 KEYWORD nonn AUTHOR Lynette O'Brien, Dec 15 2016 STATUS approved

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Last modified October 22 19:53 EDT 2019. Contains 328319 sequences. (Running on oeis4.)