A388202 a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(n+2,k).

1, 10, 79, 586, 4258, 30724, 221275, 1593946, 11494222, 83004076, 600312502, 4348191940, 31540317508, 229093439176, 1666129426963, 12131510647546, 88428766000150, 645222972915484, 4712284436622562, 34445239146428716, 251985870095247196, 1844795061256747960

Offset

0,2

Links

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

Formula

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

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

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

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

a(n) ~ 3^(-1/4) * 2^(n + 1/2) * (2 + sqrt(3))^(n + 3/2) / sqrt(Pi*n). - Vaclav Kotesovec, Sep 16 2025

D-finite with recurrence (n+2)*a(n) +2*(-7*n-6)*a(n-1) +28*(2*n-1)*a(n-2) +8*(-7*n+13)*a(n-3) +16*(n-3)*a(n-4)=0. - R. J. Mathar, Sep 16 2025

a(n) = [x^n] (1+x)^(n+2) * (3+x)^n. - Seiichi Manyama, Sep 21 2025

Mathematica

Table[Sum[ 3^k*Binomial[n, k]*Binomial[n+2, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 21 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(n+2, k));
(Magma) [&+[3^k*Binomial(n, k)*Binomial(n+2, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 21 2025

Crossrefs

Cf. A069835, A388047, A388203, A388204.

Cf. A001791, A026002, A388205.

Sequence in context: A288630 A081905 A016138 * A006329 A077245 A036732

Adjacent sequences: A388199 A388200 A388201 * A388203 A388204 A388205

Keyword

nonn

Author

Seiichi Manyama, Sep 15 2025

Status

approved