A388047 a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(n+1,k).

1, 7, 46, 307, 2086, 14374, 100108, 702979, 4968838, 35307442, 251984932, 1804984174, 12969532252, 93441190444, 674781817816, 4882868693731, 35397336684742, 257018290038586, 1868885834407732, 13607080337083642, 99187608958054516, 723792724202382292

Offset

0,2

Links

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

Formula

a(n) = [x^n] (1-2*x)^n/(1-3*x)^(n+2).

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

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

G.f.: 2/(1-8*x+4*x^2 + (1-2*x)*sqrt(1-8*x+4*x^2)).

a(n) = [x^n] (1+x)^(n+1) * (3+x)^n.

Mathematica

Table[Sum[3^k*Binomial[n, k]*Binomial[n+1, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 19 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(n+1, k));
(Magma) [&+[3^k*Binomial(n, k)*Binomial(n+1, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 19 2025

Crossrefs

Cf. A388048, A388049, A388050.

Cf. A001700, A002002, A388060, A388061.

Sequence in context: A387008 A190972 A254601 * A258340 A244265 A240722

Adjacent sequences: A388044 A388045 A388046 * A388048 A388049 A388050

Keyword

nonn

Author

Seiichi Manyama, Sep 14 2025

Status

approved