A388061 a(n) = Sum_{k=0..n} 5^k * binomial(n,k) * binomial(n+1,k).

1, 11, 106, 1011, 9726, 94526, 926836, 9153731, 90941686, 907922826, 9101393676, 91551938286, 923659741356, 9342603838236, 94709648611176, 961999228772451, 9788422857912486, 99753217183441106, 1018001237206075036, 10402025109830556026, 106410585572182121476

Offset

0,2

Links

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

Formula

a(n) = [x^n] (1-4*x)^n/(1-5*x)^(n+2).

a(n) = Sum_{k=0..n} 5^k * (-4)^(n-k) * binomial(n,k) * binomial(n+k+1,k).

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

G.f.: 2/(1-12*x+16*x^2 + (1-4*x)*sqrt(1-12*x+16*x^2)).

a(n) = [x^n] (1+x)^(n+1) * (5+x)^n.

D-finite with recurrence: (64 + 64*n)*a(n) + (-120 - 64*n)*a(1 + n) + (50 + 16*n)*a(n + 2) + (-4 - n)*a(n + 3) = 0. - Robert Israel, Feb 16 2026

Maple

f:= gfun:-rectoproc({(64 + 64*n)*a(n) + (-120 - 64*n)*a(1 + n) + (50 + 16*n)*a(n + 2) + (-4 - n)*a(n + 3), a(0) = 1, a(1) = 11, a(2) = 106}, a(n), remember):
map(f, [$0..35]); # Robert Israel, Feb 16 2026

Mathematica

Table[Sum[5^k*Binomial[n, k]*Binomial[n+1, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 18 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 5^k*binomial(n, k)*binomial(n+1, k));
(Magma) [&+[5^k*Binomial(n, k)*Binomial(n+1, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 18 2025

Crossrefs

Cf. A001700, A002002, A388047, A388060.

Sequence in context: A250416 A116011 A229070 * A365775 A058715 A140617

Adjacent sequences: A388058 A388059 A388060 * A388062 A388063 A388064

Keyword

nonn

Author

Seiichi Manyama, Sep 14 2025

Status

approved