OFFSET
0,2
COMMENTS
Partial sums of (2n)-th moment of the distance from the origin of a 3-step random walk in the plane. The subsequence of primes in this partial sum begins: 19, 751, 10847561089, 53489159252671.
LINKS
Tewodros Amdeberhan and Roberto Tauraso, Two triple binomial sum supercongruences, arXiv:1607.02483 [math.NT], Jul 08 2016.
FORMULA
a(n) = SUM[i=0..n] A002893(i) = SUM[i=0..n] SUM[p+q+r=i} (i!/(p!q!r!))^2 with p,q,r >=0.
From Sergey Perepechko Feb 16 2011: (Start)
O.g.f.: 2*sqrt(2)/Pi/(1-z)/sqrt(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))* EllipticK(8*z^(3/2)/(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))).
9*(n+2)^2*a(n) - (99+86*n+19*n^2)*a(n+1) + (72+56*n+11*n^2)*a(n+2) - (n+3)^2*a(n+3)=0.
(End)
a(n) ~ 3^(2*n + 7/2) / (32*Pi*n). - Vaclav Kotesovec, Jul 11 2016
MATHEMATICA
Accumulate[Table[Sum[Binomial[n, k]^2 Binomial[2k, k], {k, 0, n}], {n, 0, 20}]] (* Harvey P. Dale, May 05 2013 *)
PROG
(PARI) a(n)=sum(m=0, n, sum(k=0, m, binomial(m, k)^2*binomial(2*k, k)))
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Mar 08 2010
STATUS
approved