A152297 Alternate binomial partial sums of binomial(2n,n)*binomial(3n,n) (A006480).
1, 5, 79, 1427, 28447, 599435, 13100065, 293737085, 6713171455, 155700711995, 3653740285729, 86561367835805, 2067026079739921, 49689509437820933, 1201321507453119103, 29187308928225658787, 712192597620218620735
Offset: 0
Programs
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Mathematica
Table[Sum[Binomial[n,k]Binomial[2k,k]Binomial[3k,k](-1)^(n-k),{k,0,n}],{n,0,16}] -
Maxima
makelist(sum((-1)^(n-k)*binomial(n,k)*binomial(2*k,k)*binomial(3*k,k),k,0,n),n,0,16);
Formula
a(n) = sum((-1)^(n-k)*binomial(n,k)*binomial(2*k,k)*binomial(3*k,k),k=0..n).
D-finite with recurrence Recurrence: (n+3)^2*a(n+3)-(24*n^2+120*n+149)*a(n+2)-51*(n+2)^2*a(n+1)-26*(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) ~ 13*sqrt(3) * 26^n / (27*Pi*n). - Vaclav Kotesovec, Mar 02 2014