A390238 a(n) = Sum_{k=0..n} binomial(3*n+k+2,n-k).

1, 6, 38, 245, 1594, 10429, 68497, 451166, 2978230, 19694622, 130427529, 864813846, 5740250973, 38135872725, 253561166414, 1687091525061, 11232302766042, 74824855772031, 498708464207750, 3325458392854534, 22184165268292553, 148049362849110157, 988391020572075686

Offset

0,2

Links

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

Formula

G.f.: g^2/((1-3*x*g^2) * (1-x*g^4)) where g = 1+x*g^3 is the g.f. of A001764.

a(n) = Sum_{k=0..n} binomial(3*n+2,n-k) * Fibonacci(k+1).

From Vaclav Kotesovec, Nov 09 2025: (Start)

a(n) = Sum_{k=0..n} binomial(4*n-k+2, k).

a(n) ~ 3^(3*n + 5/2) / (sqrt(Pi*n) * 2^(2*n+1)). (End)

D-finite with recurrence: (2484*n^2 + 7452*n + 5520)*a(n) + (-18620*n^2 - 58636*n - 47136)*a(n + 1) + (11371*n^2 + 54803*n + 68952)*a(n + 2) + (-3093*n^2 - 20991*n - 36666)*a(n + 3) + (403*n^2 + 3377*n + 7110)*a(n + 4) + (-20*n^2 - 190*n - 450)*a(n + 5) = 0. - Robert Israel, Jul 23 2026

Maple

f:= gfun:-rectoproc({(2484*n^2 + 7452*n + 5520)*a(n) + (-18620*n^2 - 58636*n - 47136)*a(n + 1) + (11371*n^2 + 54803*n + 68952)*a(n + 2) + (-3093*n^2 - 20991*n - 36666)*a(n + 3) + (403*n^2 + 3377*n + 7110)*a(n + 4) + (-20*n^2 - 190*n - 450)*a(n + 5), a(0) = 1, a(1) = 6, a(2) = 38, a(3) = 245, a(4) = 1594}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jul 23 2026

Mathematica

a[n_]:=Sum[Binomial[3*n+k+2, n-k], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Vincenzo Librandi, Oct 31 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(3*n+k+2, n-k));
(Magma) [&+[Binomial(3*n+k+2, n-k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Oct 31 2025

Crossrefs

Cf. A004319, A087413, A390239.

Cf. A038736, A038744.

Cf. A000045, A001764.

Sequence in context: A377114 A135030 A217633 * A395655 A162558 A215466

Adjacent sequences: A390235 A390236 A390237 * A390239 A390240 A390241

Keyword

nonn,changed

Author

Seiichi Manyama, Oct 30 2025

Status

approved