A386955 a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(2*n+1,n-k).
1, 7, 42, 235, 1262, 6594, 33780, 170475, 850230, 4200130, 20585228, 100220718, 485164988, 2337145360, 11210274408, 53567616267, 255110184486, 1211287208346, 5735765695260, 27093982041546, 127699233939684, 600650635811532, 2819989050992472, 13216897613555550
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
Programs
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Magma
[&+[(k+1) * 2^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 12 2025
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Mathematica
Table[Sum[(k+1)*2^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 12 2025 *) -
PARI
a(n) = sum(k=0, n, (k+1)*2^k*binomial(2*n+1, n-k));
Formula
a(n) = [x^n] 1/((1-3*x)^2 * (1-x)^n).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n-k-1,n-k).
G.f.: 2 * (1+sqrt(1-4*x))/( sqrt(1-4*x) * (3*sqrt(1-4*x)-1)^2 ).
a(n) ~ n * 3^(2*n) / 2^(n+1). - Vaclav Kotesovec, Aug 12 2025