A385823 a(n) = Sum_{k=0..n} binomial(3*n-3,k).
1, 1, 7, 42, 256, 1586, 9949, 63004, 401930, 2579130, 16628809, 107636402, 699030226, 4552602248, 29722279084, 194458630304, 1274628824490, 8368726082346, 55027110808177, 362301656545966, 2388274575638228, 15760514137668514, 104108685843640517, 688331413734386356
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
[&+[Binomial(3*n-3,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025
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Mathematica
Table[Sum[Binomial[3*n-3,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(3*n-3, k));
Formula
a(n) = [x^n] (1+x)^(3*n-3)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-3) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-3,k) * binomial(3*n-k-4,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-4,n-k).
G.f.: 1/(g^2 * (2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.