A152297 - OEIS

%I #21 Apr 07 2022 07:16:40

%S 1,5,79,1427,28447,599435,13100065,293737085,6713171455,155700711995,

%T 3653740285729,86561367835805,2067026079739921,49689509437820933,

%U 1201321507453119103,29187308928225658787,712192597620218620735

%N Alternate binomial partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

%F a(n) = sum((-1)^(n-k)*binomial(n,k)*binomial(2*k,k)*binomial(3*k,k),k=0..n).

%F 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.

%F E.g.f.: exp(-x)*F(1/3,2/3;1,1;27*x), where F(a1,a2;b1;z) is a hypergeometric series.

%F a(n) ~ 13*sqrt(3) * 26^n / (27*Pi*n). - _Vaclav Kotesovec_, Mar 02 2014

%t Table[Sum[Binomial[n,k]Binomial[2k,k]Binomial[3k,k](-1)^(n-k),{k,0,n}],{n,0,16}]

%o (Maxima) makelist(sum((-1)^(n-k)*binomial(n,k)*binomial(2*k,k)*binomial(3*k,k),k,0,n),n,0,16);

%Y Cf. A006480, A188441, A188918, A188946.

%K nonn,easy

%O 0,2

%A _Emanuele Munarini_, Apr 14 2011