A393244 a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n,3*k).

1, 1, 1, 3, 9, 21, 47, 113, 281, 693, 1701, 4203, 10455, 26079, 65157, 163143, 409401, 1029249, 2591453, 6533759, 16494509, 41688825, 105476031, 267117585, 677069575, 1717578571, 4360390711, 11077347321, 28159605693, 71627334669, 182295199467, 464194851633

Offset

0,4

Links

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

Formula

G.f.: 1/sqrt((1-x)^2 - 4*x^3/(1-x)).

D-finite with recurrence (5 + 5*n)*a(n) + (-17 - 8*n)*a(n + 1) + (15 + 6*n)*a(n + 2) + (-13 - 4*n)*a(n + 3) + (n + 4)*a(n + 4) = 0. - Robert Israel, Feb 09 2026

a(n) ~ 5^(n + 1/2) / (2^(1/3) * sqrt(3*Pi*(21*2^(2/3) - 12*2^(1/3) - 11)*n) * (2^(4/3) - 2^(2/3) + 1)^(n-1)). - Vaclav Kotesovec, Feb 10 2026

Maple

f:= gfun:-rectoproc({(5 + 5*n)*a(n) + (-17 - 8*n)*a(n + 1) + (15 + 6*n)*a(n + 2) + (-13 - 4*n)*a(n + 3) + (n + 4)*a(n + 4), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 3}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Feb 09 2026

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n, 3*k], {k, 0, Floor[n/3]}], {n, 0, 31}] (* Vincenzo Librandi, Feb 07 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(2*k, k)*binomial(n, 3*k));
(Magma) [&+[Binomial(2*k, k)* Binomial(n, 3*k) : k in [0..Floor(n/3)]] : n in [0..35] ]; // Vincenzo Librandi, Feb 07 2026
(Python)
from math import comb
def A393244(n): return sum(comb(2*k, k) * comb(n, 3*k) for k in range(n // 3 + 1)) # Aitzaz Imtiaz, Feb 09 2026

Crossrefs

Cf. A002426, A026375, A393245.

Cf. A098479, A217615.

Cf. A360309.

Sequence in context: A000714 A380384 A267226 * A273845 A090984 A319781

Adjacent sequences: A393241 A393242 A393243 * A393245 A393246 A393247

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 07 2026

Status

approved