A382443 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^4.

1, 1, 4, 65, 566, 10912, 164032, 3237313, 62253130, 1314421886, 28392213224, 639799858304, 14785604868256, 350615631856960, 8485316740880384, 209179475361783233, 5239271305444731698, 133100429387161703962, 3424142506153260211720, 89090362800169426107070

Offset: 0

Author

Seiichi Manyama, Mar 26 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000108, A129123, A381676, A382433, A382446.

Cf. A382434.

Programs

Magma

[&+[Binomial(n, k)* (Binomial(n, k) - Binomial (n, k-1))^4: k in [0..n]]: n in [0..21]]; // Vincenzo Librandi, Mar 29 2025

Mathematica

Table[Sum[Binomial[n,k]*(Binomial[n,k]-Binomial[n,k-1])^4,{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 29 2025 *)

PARI

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^4);

Formula

a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^3.

a(n) ~ 3 * 2^(5*n+6) / (Pi^2 * 5^(5/2) * n^4). - Vaclav Kotesovec, Mar 26 2025