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, 2410285909684370788
Offset: 0
Links
- Tewodros Amdeberhan and Roberto Tauraso, Two triple binomial sum supercongruences, arXiv:1607.02483 [math.NT], Jul 08 2016.
Programs
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Mathematica
Accumulate[Table[Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}]] (* Harvey P. Dale, May 05 2013 *) -
PARI
a(n)=sum(m=0,n,sum(k=0,m,binomial(m,k)^2*binomial(2*k,k)))
Formula
a(n) = Sum_{i=0..n} A002893(i).
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