A188946 Binomial partial sums of binomial(2n,n)*binomial(3n,n) (A006480).
1, 7, 103, 1969, 41935, 947737, 22248409, 536310271, 13183283743, 328970388985, 8307368234473, 211822788505951, 5444571611722369, 140892128574440887, 3667015053678269095, 95918056089104563489, 2519845343307697266943
Offset: 0
Programs
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Magma
[&+[Binomial(n, k)*Binomial(2*k, k)*Binomial(3*k, k): k in [0..n]]: n in [0.. 18]]; // Vincenzo Librandi, Oct 16 2017
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Mathematica
Table[Sum[Binomial[n,k]Binomial[2k,k]Binomial[3k,k],{k,0,n}],{n,0,16}] Table[HypergeometricPFQ[{1/3, 2/3, -n}, {1, 1}, -27], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2017 *) -
Maxima
makelist(sum(binomial(n,k)*binomial(2*k,k)*binomial(3*k,k),k,0,n),n,0,16);
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PARI
a(n) = sum(k=0, n, binomial(n,k)*binomial(2*k,k)*binomial(3*k,k)); \\ Michel Marcus, Oct 15 2017
Formula
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*binomial(3*k,k).
Recurrence: (n+3)^2*a(n+3)-(30*n^2+150*n+187)*a(n+2)+57*(n+2)^2*a(n+1)-28*(n+1)*(n+2)*a(n)=0.
E.g.f.: exp(x)*F(1/3,2/3;1,1;27*x), where F(a1,a2;b1;z) is a hypergeometric series.
a(n) = hypergeom([1/3, 2/3, -n], [1, 1], -27). - Vladimir Reshetnikov, Oct 15 2017
a(n) ~ 2^(2*n+1) * 7^(n+1) / (3^(5/2)*Pi*n). - Vaclav Kotesovec, Nov 27 2017