A388060 a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * binomial(n+1,k).

1, 9, 73, 593, 4881, 40665, 342105, 2900385, 24742945, 212155049, 1826784873, 15785733617, 136822905649, 1189018788729, 10356383939193, 90385145624385, 790234316615745, 6919920450267465, 60682298060858505, 532817256754711953, 4683795882563500881

Offset

0,2

Links

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

Formula

a(n) = [x^n] (1-3*x)^n/(1-4*x)^(n+2).

a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(n,k) * binomial(n+k+1,k).

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

G.f.: 2/(1-10*x+9*x^2 + (1-3*x)*sqrt(1-10*x+9*x^2)).

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

D-finite with recurrence: (27 + 27*n)*a(n) + (-72 - 39*n)*a(1 + n) + (41 + 13*n)*a(n + 2) + (-4 - n)*a(n + 3) = 0. - Robert Israel, Feb 15 2026

a(n) ~ 3^(2*n+2) / (2^(3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 16 2026

Maple

f:= gfun:-rectoproc({(27 + 27*n)*a(n) + (-72 - 39*n)*a(1 + n) + (41 + 13*n)*a(n + 2) + (-4 - n)*a(n + 3), a(0) = 1, a(1) = 9, a(2) = 73}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Feb 15 2026

Mathematica

Table[Sum[4^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, 4^k*binomial(n, k)*binomial(n+1, k));
(Magma) [&+[4^k*Binomial(n, k)*Binomial(n+1, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 19 2025

Crossrefs

Cf. A001700, A002002, A388047, A388061.

Sequence in context: A023001 A277672 A015454 * A343353 A121246 A365774

Adjacent sequences: A388057 A388058 A388059 * A388061 A388062 A388063

Keyword

nonn

Author

Seiichi Manyama, Sep 14 2025

Status

approved