A390967 - OEIS

%I #27 Dec 23 2025 23:39:33

%S 0,1,18,134,738,3529,15626,66094,271592,1094970,4357916,17189832,

%T 67381030,262951065,1022931210,3970604310,15388650480,59579400030,

%U 230518577340,891563223876,3447681609924,13332176226474,51561684003588,199455373005004,771769458006512

%N a(n) = (1/(n+1)) * Sum_{k=0..n} k^4 * (k+1) * binomial(2*n-k,n-k).

%H Vincenzo Librandi, <a href="/A390967/b390967.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: x*g^7 * (1 + 11*x*g + 11*x^2*g^2 + x^3*g^3), where g = 1+x*g^2 is the g.f. of A000108.

%F a(n) = 4*n*(154*n^3+81*n^2-19*n-6)*(2*n+1)!/((n+6)!*n!). - _Tani Akinari_, Dec 23 2025

%t Table[Sum[k^4*(k+1)*Binomial[2*n-k,n-k]/(n+1),{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Dec 05 2025 *)

%o (PARI) a(n) = sum(k=0, n, k^4*(k+1)*binomial(2*n-k, n-k))/(n+1);

%o (Magma) [&+[k^4*(k+1)*Binomial(2*n-k, n-k)/(n+1): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Dec 05 2025

%o (Maxima) a(n):=4*n*(154*n^3+81*n^2-19*n-6)*(2*n+1)!/((n+6)!*n!); /* _Tani Akinari_, Dec 23 2025 */

%Y Cf. A390965, A390966.

%Y Cf. A000108, A390964.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Nov 25 2025