A305167 - OEIS

%I #26 Jun 10 2020 05:10:03

%S 0,6,198,2076,12060,49170,158418,431544,1036728,2258910,4552350,

%T 8607060,15430740,26447850,43617450,69571440,107774832,162709686,

%U 240084342,347069580,492563340,687485634,945105282,1281400104,1715452200,2269879950,2971308366

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

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

%F From _Chai Wah Wu_, Jun 09 2020: (Start)

%F 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.

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

%t 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 *)

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

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

%o (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

%Y Column 4 of A322549.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 16 2018