A387007 a(n) = Sum_{k=0..n} binomial(3*n+2,k).
1, 6, 37, 232, 1471, 9402, 60460, 390656, 2533987, 16489546, 107594213, 703680424, 4611412196, 30273024984, 199045392232, 1310535994368, 8639411571051, 57017083602138, 376674527189599, 2490742704227192, 16483857933928471, 109175823528400778, 723611538997758784
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
[&+[Binomial(3*n+2,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025
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Mathematica
Table[Sum[Binomial[3*n+2,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(3*n+2, k));
Formula
a(n) = [x^n] (1+x)^(3*n+2)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n+2) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+2,k) * binomial(3*n-k+1,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k+1,n-k).
G.f.: g^3/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(5*n+1)*(3*n-1)*(3*n-2)*a(n-2) -(295*n^3-156*n^2-61*n+6)*a(n-1) +2*n*(2*n+1)*(5*n-4)*a(n). - Georg Fischer, Aug 17 2025