A382446 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^6.

1, 1, 4, 257, 4286, 258952, 11816512, 632854273, 43732565914, 2637804065366, 207379028199080, 14568483339859880, 1205457271871693920, 95108827011788280160, 8187664948710535579904, 698818327346476962092801, 62477582066507173352034866, 5627626080883126186936773514

Offset: 0

Author

Seiichi Manyama, Mar 26 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A000108, A129123, A381676, A382433, A382443.

Cf. A382435.

Programs

Magma

[&+[Binomial(n, k)* (Binomial(n, k) - Binomial (n, k-1))^6: k in [0..n]]: n in [0..21]]; // Vincenzo Librandi, Mar 30 2025

Mathematica

Table[Sum[Binomial[n,k]*(Binomial[n,k]-Binomial[n,k-1])^6,{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 30 2025 *)

PARI

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^6);

Formula

a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^5.

a(n) ~ 15 * 2^(7*n+9) / (Pi^3 * 7^(7/2) * n^6). - Vaclav Kotesovec, Mar 26 2025