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

%I A305167 #26 Jun 10 2020 05:10:03
%S A305167 0,6,198,2076,12060,49170,158418,431544,1036728,2258910,4552350,
%T A305167 8607060,15430740,26447850,43617450,69571440,107774832,162709686,
%U A305167 240084342,347069580,492563340,687485634,945105282,1281400104,1715452200,2269879950,2971308366
%N A305167 Constant term in the expansion of (Sum_{k=0..n} k*(x^k + x^(-k)))^4.
%F A305167 a(n) = n * (n+1) * (2*n+1) * (17*n^4+34*n^3+28*n^2+11*n+15) / 105.
%F A305167 From _Chai Wah Wu_, Jun 09 2020: (Start)
%F A305167 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 A305167 G.f.: 6*x*(x + 1)*(x^4 + 24*x^3 + 86*x^2 + 24*x + 1)/(x - 1)^8. (End)
%t A305167 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 A305167 (PARI) {a(n) = polcoeff((sum(k=0, n, k*(x^k+x^(-k))))^4, 0, x)}
%o A305167 (PARI) {a(n) = n*(n+1)*(2*n+1)*(17*n^4+34*n^3+28*n^2+11*n+15)/105}
%o A305167 (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 A305167 Column 4 of A322549.
%K A305167 nonn
%O A305167 0,2
%A A305167 _Seiichi Manyama_, Dec 16 2018