A318119 Constant term in the expansion of (Sum_{k=0..n} k*(x^k + x^(-k)))^5.

0, 0, 560, 14240, 146680, 922680, 4226040, 15492680, 48144680, 131678360, 325322360, 739761880, 1570082800, 3143824320, 5988841040, 10926565040, 19197225520, 32624627920, 53829216160, 86499340720, 135731931720, 208455129960, 313946860040, 464464838200

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Formula

a(n) = (n-1) * n * (n+1) * (2*n+1) * (967*n^5+3868*n^4+6005*n^3+5444*n^2+2844*n+1008) / 9072.

Mathematica

a[n_] := Coefficient[Expand[Sum[k * (x^k + x^(-k)), {k, 0, n}]^5], x, 0]; Array[a, 30, 0] (* Amiram Eldar, Dec 16 2018 *)
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 0, 560, 14240, 146680, 922680, 4226040, 15492680, 48144680, 131678360}, 40] (* Harvey P. Dale, Jun 16 2025 *)

Prog

(PARI) {a(n) = polcoeff((sum(k=0, n, k*(x^k+x^(-k))))^5, 0, x)}
(PARI) {a(n) = (n-1)*n*(n+1)*(2*n+1)*(967*n^5+3868*n^4+6005*n^3+5444*n^2+2844*n+1008)/9072}
(GAP) List([0..25], n->(n-1)*n*(n+1)*(2*n+1)*(967*n^5+3868*n^4+6005*n^3+5444*n^2+2844*n+1008)/9072); # Muniru A Asiru, Dec 16 2018

Crossrefs

Column 5 of A322549.

Cf. A083669.

Sequence in context: A069243 A169719 A104591 * A364151 A196568 A171347

Adjacent sequences: A318116 A318117 A318118 * A318120 A318121 A318122

Keyword

nonn,easy

Author

Seiichi Manyama, Dec 16 2018

Status

approved