A377011 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+1,n-k).
1, 6, 34, 188, 1026, 5556, 29940, 160824, 862018, 4613636, 24667644, 131795912, 703812916, 3757135752, 20051429544, 106992663408, 570827898306, 3045193326372, 16244056119084, 86646747723048, 462161936699196, 2465043081687192, 13147597801986264, 70123266087502608
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
-
Magma
[&+[3^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
-
Mathematica
Table[Sum[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, 3^k*binomial(2*n+1, n-k));
Formula
a(n) = [x^n] 1/((1-4*x) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 4^k * binomial(2*n-k,n-k).
G.f.: 1/( sqrt(1-4*x) * (2*sqrt(1-4*x)-1) ).
a(n) ~ 2^(4*n+2) / 3^(n+1). - Vaclav Kotesovec, Aug 20 2025
D-finite with recurrence 3*n*a(n) +2*(-14*n+3)*a(n-1) +32*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 21 2025