A387008 a(n) = Sum_{k=0..n} binomial(3*n+3,k).
1, 7, 46, 299, 1941, 12616, 82160, 536155, 3505699, 22964087, 150676186, 990134948, 6515349244, 42925973608, 283134975936, 1869455684187, 12355133446527, 81725384344741, 541021064605298, 3584203906519219, 23761237400402597, 157623924396214756, 1046244086051121248
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}] (* Vaclav Kotesovec, Aug 20 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+2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k+2,n-k).
G.f.: g^4/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(3*n-2)*(3*n-1)*(5*n^2+n-2)*a(n-2) -(295*n^4-156*n^3-339*n^2+12*n+20)*a(n-1) +2*(2*n+1)*(n+1)*(5*n^2-9*n+2)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 3^(3*n + 7/2) / (sqrt(Pi*n) * 2^(2*n+3)). - Vaclav Kotesovec, Aug 20 2025