A378031 Cogrowth sequence for the 18-element group C6 X C3 = <S,T | S^6, T^3, [S,T]>.

1, 1, 2, 85, 926, 5461, 37130, 349525, 2973350, 22369621, 174174002, 1431655765, 11582386286, 91625968981, 729520967450, 5864062014805, 47006639297270, 375299968947541, 2999857885752002, 24019198012642645, 192222214478506046, 1537228672809129301

Offset

0,3

Comments

Sequence gives terms for n = 0 (mod 3), all other terms are 0.

Links

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

Index entries for linear recurrences with constant coefficients, signature (7,-19,189,216).

Formula

G.f.: (36*x^4+99*x^3-14*x^2+6*x-1) / ((8*x-1) * (x+1) * (27*x^2+1)).

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n,6*k). - Seiichi Manyama, Sep 10 2025

Mathematica

CoefficientList[Series[(36*x^4+99*x^3-14*x^2+6*x-1)/((8*x-1)*(x+1)*(27*x^2+1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 09 2025 *)

Prog

(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (36*x^4+99*x^3-14*x^2+6*x-1) / ((8*x-1) * (x+1) * (27*x^2+1)))); // Vincenzo Librandi, Oct 09 2025

Crossrefs

Cf. A095364 (D9), A377627 (C6 X C2), A007613 (C3 X C3), A378109 (S3 X C3), A378110 (S3:C3).

Sequence in context: A078166 A101578 A041881 * A076542 A226842 A303892

Adjacent sequences: A378028 A378029 A378030 * A378032 A378033 A378034

Keyword

nonn,easy

Author

Sean A. Irvine, Nov 14 2024

Status

approved