A392270 a(n) = Sum_{k=0..floor(2*n/5)} binomial(k+2,2) * binomial(k,2*n-5*k).

1, 0, 0, 3, 0, 6, 6, 0, 30, 10, 15, 90, 15, 105, 210, 49, 420, 420, 280, 1260, 801, 1296, 3150, 1755, 4665, 6996, 4950, 13915, 14718, 15840, 36193, 31746, 49335, 85527, 75075, 141261, 191191, 195195, 370590, 422345, 528681, 904440, 959260, 1413771, 2099670

Offset

0,4

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,3,-15,0,-12,20,-3,18,-15,6,-12,7,-3,3,-1).

Formula

G.f.: (1-x^3) * ((1-x^3)^2 + 3*x^5) / ((1-x^3)^2 - x^5)^3.

a(n) = 6*a(n-3) + 3*a(n-5) - 15*a(n-6) - 12*a(n-8) + 20*a(n-9) - 3*a(n-10) + 18*a(n-11) - 15*a(n-12) + 6*a(n-13) - 12*a(n-14) + 7*a(n-15) - 3*a(n-16) + 3*a(n-17) - a(n-18).

Mathematica

CoefficientList[Series[(1-x^3)*((1-x^3)^2+3*x^5)/((1-x^3)^2-x^5)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jan 07 2026 *)

Prog

(PARI) my(N=50, x='x+O('x^N)); Vec((1-x^3)*((1-x^3)^2+3*x^5)/((1-x^3)^2-x^5)^3)
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-x^3) * ((1-x^3)^2 + 3*x^5) / ((1-x^3)^2 - x^5)^3); // Vincenzo Librandi, Jan 07 2026

Crossrefs

Cf. A391963, A392252, A392269.

Sequence in context: A199610 A285871 A382340 * A322579 A004606 A019808

Adjacent sequences: A392267 A392268 A392269 * A392271 A392272 A392273

Keyword

nonn

Author

Seiichi Manyama, Jan 05 2026

Status

approved