OFFSET
0,1
COMMENTS
This sequence is s_18 in Cooper's paper.
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Every prime eventually divides some term of this sequence. - Amita Malik, Aug 20 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..830 (terms 0..254 from Jason Kimberley)
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See s18 p. 3.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
FORMULA
1/Pi
= 2*3^(-5/2) Sum {k>=0} (n a(n)/18^n) [Cooper, equation (42)]
= 2*3^(-5/2) Sum {k>=0} (n a(n)/A001027(n)).
G.f.: 1+hypergeom([1/8, 3/8],[1],256*x^3/(1-12*x)^2)^2/sqrt(1-12*x). - Mark van Hoeij, May 07 2013
Conjecture D-finite with recurrence: n^3*a(n) -2*(2*n-1)*(7*n^2-7*n+3)*a(n-1) +12*(4*n-5)*(n-1)* (4*n-3)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
a(n) ~ 3 * 2^(4*n + 1/2) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023
MATHEMATICA
Table[Sum[(-1)^j*Binomial[n, j]*Binomial[2j, j]*Binomial[2n-2j, n-j]* (Binomial[2n-3j-1, n] +Binomial[2n-3j, n]), {j, 0, Floor[n/3]}], {n, 0, 20}] (* G. C. Greubel, Oct 24 2017 *)
PROG
(Magma) s_18 := func<k|&+[(-1)^j*C(k, j)*C(2*j, j)*C(2*k-2*j, k-j)*(C(2*k-3*j-1, k)+C(2*k-3*j, k)):j in[0..k div 3]]> where C is Binomial;
(PARI) {a(n) = sum(j=0, floor(n/3), (-1)^j*binomial(n, j)*binomial(2*j, j)* binomial(2*n-2*j, n-j)*(binomial(2*n-3*j-1, n) +binomial(2*n-3*j, n)))}; \\ G. C. Greubel, Apr 02 2019
(Sage) [sum((-1)^j*binomial(n, j)*binomial(2*j, j)*binomial(2*n-2*j, n-j)* (binomial(2*n-3*j-1, n)+binomial(2*n-3*j, n)) for j in (0..floor(n/3))) for n in (0..20)] # G. C. Greubel, Apr 02 2019
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,easy
AUTHOR
Jason Kimberley, Nov 25 2012
STATUS
approved