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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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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
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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 *) -
PARI
a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(2*n-k, n-k));
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).