A386942 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(2*n-k,n-k).

1, 8, 54, 340, 2060, 12180, 70812, 406656, 2313630, 13067340, 73372728, 410013864, 2282066332, 12658839200, 70017730680, 386314361808, 2126818591932, 11686657363236, 64108376373700, 351142219736000, 1920711937207140, 10493241496749000, 57263080117042800

Offset

0,2

Links

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

Formula

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

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

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

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

Mathematica

Table[Sum[(2*k+1) *Binomial[2*k, k]* Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 03 2025 *)

Prog

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

Crossrefs

Cf. A293490, A377011, A383832.

Cf. A386940, A386941.

Sequence in context: A081899 A057970 A208310 * A154235 A289796 A287814

Adjacent sequences: A386939 A386940 A386941 * A386943 A386944 A386945

Keyword

nonn,easy

Author

Seiichi Manyama, Aug 10 2025

Status

approved