A052784 Expansion of e.g.f.: x^3*(exp(x)-1)^3.

0, 0, 0, 0, 0, 0, 720, 7560, 50400, 272160, 1300320, 5738040, 23958000, 96061680, 373477104, 1417295880, 5274360000, 19313985600, 69770966976, 249130574424, 880629138000, 3085339798800, 10724762719440, 37017144571176, 126954125892000, 432884978460000

Offset

0,7

Comments

Previous name was: A simple grammar.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 741

Formula

E.g.f.: x^3*exp(x)^3-3*x^3*exp(x)^2+3*x^3*exp(x)-x^3 = x^3*(exp(x)-1)^3.

D-finite Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, (-36*n^2-66*n-6*n^3-36)*a(n)+(-44*n+11*n^3+33*n^2-132)*a(n+1)+(-6*n^3-36+42*n)*a(n+2)+(-3*n^2+n^3+2*n)*a(n+3)=0}

For n>3 , a(n) = (n-2)*(n-1)*n*(3^(n-3) - 3*2^(n-3) + 3). - Vaclav Kotesovec, Oct 01 2013

Maple

spec := [S, {B=Set(Z, 1 <= card), S=Prod(Z, Z, Z, B, B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

CoefficientList[Series[x^3*(E^x-1)^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

Prog

(PARI) seq(n)=Vec(serlaplace(x^3*(exp(x + O(x^(n-4))) - 1)^3), -n-1) \\ Andrew Howroyd, Oct 29 2025

Crossrefs

Sequence in context: A153760 A029574 A119540 * A052786 A187192 A052792

Adjacent sequences: A052781 A052782 A052783 * A052785 A052786 A052787

Keyword

easy,nonn

Author

INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000

Extensions

New name, using e.g.f., by Vaclav Kotesovec, Oct 01 2013

a(21) onward from Andrew Howroyd, Oct 29 2025

Status

approved