A387033 a(n) = Sum_{k=0..n} binomial(3*n-1,k).

1, 3, 16, 93, 562, 3473, 21778, 137980, 880970, 5658537, 36519556, 236618693, 1538132224, 10026362492, 65513177704, 428957009288, 2813768603466, 18486790962201, 121634649321208, 801330506737399, 5285305708097522, 34896814868837161, 230631268849574378

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = [x^n] (1+x)^(3*n-1)/(1-x).

a(n) = [x^n] 1/((1-x)^(2*n-1) * (1-2*x)).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-1,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.: 1/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.

D-finite with recurrence: 24*(5*n+6)*(3*n-4)*(3*n-5)*a(n-2)-(295*n^3-451*n^2-234*n+360)*a(n-1)+2*n*(5*n+1)*(2*n-3)*a(n) = 0. - Georg Fischer, Aug 17 2025

a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi*n) * 2^(2*n-1)). - Vaclav Kotesovec, Aug 27 2025

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n,n-2*k). - Seiichi Manyama, Nov 11 2025

Mathematica

Table[Sum[Binomial[3*n-1, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 27 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(3*n-1, k));
(Magma) [&+[Binomial(3*n-1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Crossrefs

Cf. A066380, A160906, A385823, A386006, A387007, A387008.

Cf. A114121, A387037.

Cf. A001764, A047099, A165817.

Sequence in context: A305852 A221764 A371742 * A213229 A323968 A074555

Adjacent sequences: A387030 A387031 A387032 * A387034 A387035 A387036

Keyword

nonn

Author

Seiichi Manyama, Aug 13 2025

Status

approved