A383832 a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n+2,n-k).
1, 10, 78, 548, 3630, 23148, 143724, 874888, 5245038, 31065500, 182189348, 1059775608, 6122246572, 35160205752, 200902089240, 1142857957392, 6475994731758, 36569545322364, 205869970843764, 1155749458070040, 6472151016349284, 36161680227612456, 201628061114911848
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
[&+[(k+1) * 3^k * Binomial(2*n+2,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
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Mathematica
Table[Sum[(k+1)* 3^k * Binomial[2*n+2,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *) -
PARI
a(n) = sum(k=0, n, (k+1)*3^k*binomial(2*n+2, n-k));
Formula
a(n) = [x^n] 1/((1-4*x)^2 * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+2,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} (k+1) * 4^k * binomial(2*n-k,n-k).
G.f.: 1/( sqrt(1-4*x) * (2*sqrt(1-4*x)-1)^2 ).
D-finite with recurrence 15*n*a(n) +2*(-94*n+23)*a(n-1) +192*(4*n-3)*a(n-2) +512*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Aug 21 2025