A174123 Partial sums of A002893.

1, 4, 19, 112, 751, 5404, 40573, 313408, 2471167, 19791004, 160459069, 1313922064, 10847561089, 90174127684, 754009158019, 6336733626112, 53489159252671, 453258909448636, 3854034482891725, 32871004555812112, 281127047928811201

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

Table of n, a(n) for n=0..20.

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

Cf. A002893, A000172, A002895, A000984.

Sequence in context: A243241 A060905 A304473 * A127548 A331326 A122835

Adjacent sequences: A174120 A174121 A174122 * A174124 A174125 A174126

Keyword

easy,nonn

Author

Jonathan Vos Post, Mar 08 2010

Status

approved