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[0] == 1}, a, {n, 0, 50}] (* G. C. Greubel, Jul 04 2018 *)
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 *)
PROG
(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
CROSSREFS
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.)
KEYWORD
nonn
AUTHOR
Lynette O'Brien, Dec 15 2016
STATUS
approved