A388410 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n,k) * binomial(n+3,k).

1, 7, 49, 344, 2426, 17189, 122313, 873660, 6261150, 45001390, 324266842, 2341813984, 16945884500, 122841115505, 891888498425, 6484788936276, 47210685047862, 344106251099682, 2510775437779582, 18337791689669488, 134052696223395436, 980760386365387442

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = [x^n] (1+2*x)^n/(1-x)^(n+4).

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

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

G.f.: 1/(sqrt(1-8*x+4*x^2) * ((1+2*x + sqrt(1-8*x+4*x^2))/2)^3).

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

Mathematica

Table[Sum[ 3^(n-k)*Binomial[ n, k]*Binomial[n+3, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 21 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n, k)*binomial(n+3, k));
(Magma) [&+[3^(n-k)*Binomial(n, k)*Binomial(n+3, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 21 2025

Crossrefs

Cf. A098663, A388409, A388411.

Cf. A388203.

Sequence in context: A050737 A195908 A033143 * A024582 A024587 A351057

Adjacent sequences: A388407 A388408 A388409 * A388411 A388412 A388413

Keyword

nonn

Author

Seiichi Manyama, Sep 17 2025

Status

approved