A328735 Constant term in the expansion of (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.

1, 0, 14, 72, 882, 8400, 95180, 1060080, 12389650, 146472480, 1767391164, 21581516880, 266718438756, 3327025429728, 41849031952728, 530135326392672, 6757845419895570, 86619827323917888, 1115719258312182524, 14434274832755201424, 187477238295444829732

Offset: 0

Author

Seiichi Manyama, Oct 26 2019

Keywords

• nonn

Crossrefs

Column k=4 of A328748.

Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^m: A126869 (m=2), A002898 (m=3), this sequence (m=4), A328751 (m=5).

Cf. A002899, A005260, A328725.

Programs

Mathematica

Table[Sum[(-2)^(n-i)*Binomial[n,i] * Sum[Binomial[i,j]^4, {j,0,i}], {i,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 20 2023 *)

PARI

{a(n) = polcoef(polcoef(polcoef((-2+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}

PARI

{a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

Formula

a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.

From Vaclav Kotesovec, Mar 20 2023: (Start)

Recurrence: n^3*a(n) = 2*(n-1)*n*(2*n - 1)*a(n-1) + 112*(n-1)^3*a(n-2) + 184*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 336*(n-3)*(n-2)*(n-1)*a(n-4).

a(n) ~ 2^(n-4) * 7^(n + 3/2) / (Pi^(3/2) * n^(3/2)). (End)