A377150 a(n) = Sum_{k=0..floor(n/3)} binomial(k+3,3) * binomial(k,n-3*k)^2.

1, 0, 0, 4, 4, 0, 10, 40, 10, 20, 180, 180, 55, 560, 1260, 616, 1435, 5600, 5684, 4424, 18956, 33720, 24780, 55944, 147249, 157560, 182280, 523540, 826440, 802560, 1681966, 3531880, 4072035, 5671084, 12941764, 19281064, 22523175, 43823520, 80254746, 99744776

Offset: 0

Author

Seiichi Manyama, Oct 18 2024

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..5705

Crossrefs

Cf. A377148, A377149.

Cf. A089627.

Programs

Maple

f:= proc(n) local k; add(binomial(k+3,3)*binomial(k,n-3*k)^2,k=0..floor(n/3)) end proc:
map(f, [$0..50]); # Robert Israel, Dec 05 2024

PARI

a(n) = sum(k=0, n\3, binomial(k+3, 3)*binomial(k, n-3*k)^2);

PARI

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=3, M=40, x='x+O('x^M), X=1-x^3-x^4, Y=7); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

Formula

G.f.: (1-x^3-x^4) * ((1-x^3-x^4)^2 + 6*x^7) / ((1-x^3-x^4)^2 - 4*x^7)^(7/2).