A328809 Constant term in the expansion of (1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

1, 3, 39, 597, 11991, 260613, 6129489, 151078707, 3867441111, 101852866533, 2744610170049, 75348380209347, 2100889194001761, 59349600029522403, 1695505948476461559, 48909452234258070117, 1422877722974198091351, 41704912707174877940613

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.

From Vaclav Kotesovec, Oct 28 2019: (Start)

Recurrence: n^4*(40*n^2 - 24*n - 79)*a(n) = (1080*n^6 - 2808*n^5 + 875*n^4 + 2928*n^3 - 3762*n^2 + 1834*n - 336)*a(n-1) + (9320*n^6 - 42872*n^5 + 61193*n^4 - 12152*n^3 - 35518*n^2 + 21658*n - 2016)*a(n-2) - (n-2)*(48560*n^5 - 223376*n^4 + 216118*n^3 + 381866*n^2 - 791133*n + 355194)*a(n-3) + (n-3)*(n-2)*(79560*n^4 - 286416*n^3 - 56675*n^2 + 976675*n - 616322)*a(n-4) - 11*(n-4)*(n-3)*(n-2)*(5080*n^3 - 8128*n^2 - 25641*n + 21693)*a(n-5) + 363*(n-5)*(n-4)*(n-3)*(n-2)*(40*n^2 + 56*n - 63)*a(n-6).

a(n) ~ 33^(n+2) / (256 * sqrt(5) * Pi^2 * n^2). (End)

Mathematica

Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^5, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)

Prog

(PARI) {a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}

Crossrefs

Column k=5 of A328807.

Cf. A005261, A328750, A328751.

Sequence in context: A361539 A014850 A341671 * A327603 A336540 A228749

Adjacent sequences: A328806 A328807 A328808 * A328810 A328811 A328812

Keyword

nonn

Author

Seiichi Manyama, Oct 28 2019

Status

approved