A377011 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+1,n-k).

1, 6, 34, 188, 1026, 5556, 29940, 160824, 862018, 4613636, 24667644, 131795912, 703812916, 3757135752, 20051429544, 106992663408, 570827898306, 3045193326372, 16244056119084, 86646747723048, 462161936699196, 2465043081687192, 13147597801986264, 70123266087502608

Offset

0,2

Links

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

Formula

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

a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).

a(n) = Sum_{k=0..n} 4^k * binomial(2*n-k,n-k).

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

a(n) ~ 2^(4*n+2) / 3^(n+1). - Vaclav Kotesovec, Aug 20 2025

D-finite with recurrence 3*n*a(n) +2*(-14*n+3)*a(n-1) +32*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 21 2025

a(n) = binomial(1+2*n, n)*hypergeom([1, -n], [2+n], -3). - Stefano Spezia, Nov 05 2025

Mathematica

Table[Sum[3^k * Binomial[2*n+1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 03 2025 *)
a[n_]:=Binomial[1+2 n, n]*Hypergeometric2F1[1, -n, 2+n, -3]; Array[a, 24, 0] (* Stefano Spezia, Nov 05 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^k*binomial(2*n+1, n-k));
(Magma) [&+[3^k * Binomial(2*n+1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025

Crossrefs

Cf. A293490, A383832, A386942.

Cf. A000302, A000984, A026641, A141223, A386957.

Sequence in context: A229009 A085351 A125343 * A163350 A320746 A320749

Adjacent sequences: A377008 A377009 A377010 * A377012 A377013 A377014

Keyword

nonn

Author

Seiichi Manyama, Aug 11 2025

Status

approved