A388053 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2+k+1,n).

1, 5, 61, 1561, 63241, 3522221, 249612805, 21462603121, 2168625812497, 251675182111573, 32975850501891661, 4813410306776846985, 774409806857044768281, 136129335881578036699261, 25955449985407590076305941, 5334819874165654299168927201, 1175777209805774164775256809505

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = [x^n] (1-x)^n/(1-2*x)^(n^2+2).

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

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

a(n) = [x^n] (1+x)^(n^2+1) * (2+x)^n.

Mathematica

Table[Sum[Binomial[n, k]*Binomial[n^2+k+1, n], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 19 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n^2+k+1, n));
(Magma) [&+[Binomial(n, k)*Binomial(n^2+k+1, n): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 19 2025

Crossrefs

Main diagonal of A388052.

Cf. A306280.

Sequence in context: A000364 A393752 A159316 * A361556 A231798 A258672

Adjacent sequences: A388050 A388051 A388052 * A388054 A388055 A388056

Keyword

nonn

Author

Seiichi Manyama, Sep 14 2025

Status

approved