A305167 Constant term in the expansion of (Sum_{k=0..n} k*(x^k + x^(-k)))^4.

0, 6, 198, 2076, 12060, 49170, 158418, 431544, 1036728, 2258910, 4552350, 8607060, 15430740, 26447850, 43617450, 69571440, 107774832, 162709686, 240084342, 347069580, 492563340, 687485634, 945105282, 1281400104, 1715452200, 2269879950, 2971308366

Offset: 0

Author

Seiichi Manyama, Dec 16 2018

Keywords

• nonn

Crossrefs

Column 4 of A322549.

Programs

GAP

List([0..30],n->n*(n+1)*(2*n+1)*(17*n^4+34*n^3+28*n^2+11*n+15)/105); # Muniru A Asiru, Dec 16 2018

Mathematica

a[n_] := Coefficient[Expand[Sum[k * (x^k + x^(-k)), {k, 0, n}]^4], x, 0]; Array[a, 30, 0] (* Amiram Eldar, Dec 16 2018 *)

PARI

{a(n) = polcoeff((sum(k=0, n, k*(x^k+x^(-k))))^4, 0, x)}

PARI

{a(n) = n*(n+1)*(2*n+1)*(17*n^4+34*n^3+28*n^2+11*n+15)/105}

Formula

a(n) = n * (n+1) * (2*n+1) * (17*n^4+34*n^3+28*n^2+11*n+15) / 105.

From Chai Wah Wu, Jun 09 2020: (Start)

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n > 7.

G.f.: 6*x*(x + 1)*(x^4 + 24*x^3 + 86*x^2 + 24*x + 1)/(x - 1)^8. (End)