A382446 -id:A382446 - cp's OEIS frontend

A381676 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^2.

1, 1, 4, 17, 86, 472, 2752, 16753, 105394, 680366, 4484360, 30067160, 204508240, 1408057120, 9796738304, 68786005361, 486845236106, 3470187822754, 24891491746792, 179556655434382, 1301857088258836, 9482632068303296, 69361538748381824, 509303099950899352

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A129123, A382433, A382443, A382446.
Cf. A131428, A178824.
Cf. A000172, A086618, A238115.

[ &+[Binomial(n, k)^2 * (Binomial(n, k) - (k gt 0 select Binomial(n, k-1) else 0)) : k in [0..n]] : n in [0..20] ]; // Vincenzo Librandi, Mar 27 2025

Table[Sum[Binomial[n,k]^2*(Binomial[n,k]-Binomial[n,k-1]),{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 27 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^2);

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

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

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Mar 26 2025

nonn

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

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

Cf. A382434.

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

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

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

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

cp's OEIS Frontend

A381676 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^2.

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