A386956 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+1,n-k).
1, 19, 282, 3763, 47294, 571950, 6733668, 77723187, 883589238, 9924844474, 110396411372, 1218075749934, 13348677037868, 145438914042172, 1576690043132376, 17018212213758771, 182983432175308710, 1960781840268630786, 20947171352106580284, 223169444039365834362
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
[&+[(k+1) * 8^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
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Mathematica
Table[Sum[(k+1) * 8^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *) -
PARI
a(n) = sum(k=0, n, (k+1)*8^k*binomial(2*n+1, n-k));
Formula
a(n) = [x^n] 1/((1-9*x)^2 * (1-x)^n).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} (k+1) * 9^k * binomial(2*n-k-1,n-k).
G.f.: 2 * (1+sqrt(1-4*x))/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7)^2 ).