A328808 - OEIS

%I #11 Oct 28 2019 05:31:50

%S 1,3,23,225,2583,32133,422069,5757699,80790775,1158593589,16905540753,

%T 250185539079,3746205581589,56652844671855,864032059578879,

%U 13274539401672345,205252378269637815,3191578469685269925,49876569284504593505,782943268394316187815

%N Constant term in the expansion of (3 + x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.

%F a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.

%F From _Vaclav Kotesovec_, Oct 28 2019: (Start)

%F Recurrence: n^3*a(n) = (2*n - 1)*(8*n^2 - 8*n + 3)*a(n-1) + (n-1)*(22*n^2 - 44*n + 13)*a(n-2) - 44*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 51*(n-3)*(n-2)*(n-1)*a(n-4).

%F a(n) ~ sqrt(2) * 17^(n + 3/2) / (64 * Pi^(3/2) * n^(3/2)). (End)

%t Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^4, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 28 2019 *)

%o (PARI) {a(n) = polcoef(polcoef(polcoef((1+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}

%o (PARI) {a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

%Y Column k=4 of A328807.

%Y Cf. A005260, A328725, A328735.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 28 2019