A392316 a(n) = Sum_{k=0..floor(3*n/8)} binomial(k+2,2) * binomial(k,3*n-8*k).

1, 0, 0, 3, 0, 0, 6, 0, 10, 10, 0, 60, 15, 0, 210, 21, 28, 560, 28, 252, 1260, 36, 1260, 2520, 100, 4620, 4620, 715, 13860, 7920, 4356, 36036, 12961, 20098, 84084, 21385, 75166, 180180, 40950, 240345, 360496, 105560, 680800, 683128, 340340, 1750456, 1248480

Offset

0,4

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,0,9,0,0,-36,0,3,84,0,-18,-126,0,45,126,-3,-60,-84,9,45,36,-9,-18,-8,3,3,1).

Formula

G.f.: ((1-x^3)^6 + 7*x^8*(1-x^3)^3 + x^16) / ((1-x^3)^3 - x^8)^3.

a(n) = 9*a(n-3) - 36*a(n-6) + 3*a(n-8) + 84*a(n-9) - 18*a(n-11) - 126*a(n-12) + 45*a(n-14) + 126*a(n-15) - 3*a(n-16) - 60*a(n-17) - 84*a(n-18) + 9*a(n-19) + 45*a(n-20) + 36*a(n-21) - 9*a(n-22) - 18*a(n-23) - 8*a(n-24) + 3*a(n-25) + 3*a(n-26) + a(n-27).

Mathematica

CoefficientList[Series[((1-x^3)^6+7*x^8*(1-x^3)^3+x^16)/((1-x^3)^3-x^8)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jan 16 2026 *)

Prog

(PARI) my(N=50, x='x+O('x^N)); Vec(((1-x^3)^6+7*x^8*(1-x^3)^3+x^16)/((1-x^3)^3-x^8)^3)
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R! ((1-x^3)^6 + 7*x^8*(1-x^3)^3 + x^16) / ((1-x^3)^3 - x^8)^3); // Vincenzo Librandi, Jan 16 2026

Crossrefs

Cf. A392255, A392313, A392314.

Sequence in context: A390036 A321429 A145225 * A332442 A061480 A220692

Adjacent sequences: A392313 A392314 A392315 * A392317 A392318 A392319

Keyword

nonn,easy

Author

Seiichi Manyama, Jan 06 2026

Status

approved