A384365 a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n+1,n-k).
1, 9, 67, 458, 2979, 18750, 115278, 696372, 4149283, 24452534, 142808922, 827780684, 4767638158, 27309438252, 155689424316, 883891633896, 4999703023395, 28188457323366, 158463492162594, 888473780483292, 4969653746436762, 27737520941131140, 154507945286680452
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+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) * 3^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)*3^k*binomial(2*n+1, n-k));
Formula
a(n) = [x^n] 1/((1-4*x)^2 * (1-x)^n).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} (k+1) * 4^k * binomial(2*n-k-1,n-k).
G.f.: (1+sqrt(1-4*x))/( 2 * sqrt(1-4*x) * (2*sqrt(1-4*x)-1)^2 ).
D-finite with recurrence +27*n*a(n) +6*(-58*n+17)*a(n-1) +32*(46*n-37)*a(n-2) +1024*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Aug 19 2025
a(n) ~ n * 2^(4*n+1) / 3^(n+1). - Vaclav Kotesovec, Aug 20 2025