A392033 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(n,k) * binomial(n-k,k).

1, 1, 9, 25, 109, 381, 1421, 5069, 18075, 63499, 221323, 764523, 2621587, 8928115, 30220347, 101721707, 340659227, 1135543323, 3769023795, 12460643043, 41045661279, 134748546639, 440974812831, 1438893866655, 4682226213021, 15197079464301, 49206276341901

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: ((1-x)^6 - 6*x^2*(1-x)^4 + 18*x^4*(1-x)^2 - 20*x^6) / ((1-x)^2 - 4*x^2)^(7/2).

Mathematica

CoefficientList[Series[((1-x)^6-6*x^2*(1-x)^4+18*x^4*(1-x)^2-20*x^6)/((1-x)^2-4*x^2)^(7/2), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 31 2025 *)

Prog

(PARI) a098473(n, k) = binomial(n, k)*binomial(2*k, k);
my(A=1, B=1, C=A*B, N=3, M=30, x='x+O('x^M), X=1-B*x, Y=2); Vec(sum(k=0, N, (-C)^k*a098473(N, k)*X^(2*N-2*k)*x^(Y*k))/(X^2-4*C*x^Y)^(N+1/2))
(Magma) m := 50; R<x> := PowerSeriesRing(RationalField(), m); Coefficients(((1-x)^6 - 6*x^2*(1-x)^4 + 18*x^4*(1-x)^2 - 20*x^6) / ((1-x)^2 - 4*x^2)^(7/2)); // Vincenzo Librandi, Dec 31 2025

Crossrefs

Cf. A002426, A391990, A392032.

Cf. A098473.

Sequence in context: A383077 A083672 A193644 * A083100 A084058 A108570

Adjacent sequences: A392030 A392031 A392032 * A392034 A392035 A392036

Keyword

nonn

Author

Seiichi Manyama, Dec 27 2025

Status

approved