A052777 E.g.f.: x^2*(exp(x)-1)^3.

0, 0, 0, 0, 0, 120, 1080, 6300, 30240, 130032, 521640, 1996500, 7389360, 26676936, 94486392, 329647500, 1136116800, 3876164832, 13112135496, 44031456900, 146920942800, 487489214520, 1609441068312, 5289755245500, 17315399138400, 56470807803600, 183546483143400

Offset

0,6

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 734

Index entries for linear recurrences with constant coefficients, signature (18,-141,630,-1767,3222,-3815,2826,-1188,216).

Formula

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

Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (-36*n^2-66*n-6*n^3-36)*a(n)+(11*n+11*n^3+44*n^2-66)*a(n+1)+(-12*n^2+18*n-6*n^3)*a(n+2)+(n^3-n)*a(n+3), a(5)=120}.

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

a(n) = n*A052761(n-1) = 3!*n*(n-1)*Stirling2(n-2,3) for n >= 2. - Andrew Howroyd, Aug 08 2020

Maple

spec := [S, {B=Set(Z, 1 <= card), S=Prod(Z, Z, B, B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20); # end of program
seq(6*(n^2-n)*combinat[stirling2](n-2, 3), n=0..20);  # Mark van Hoeij, May 29 2013

Mathematica

CoefficientList[Series[x^2*(E^x-1)^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

Prog

(PARI) x='x+O('x^66); concat([0, 0, 0, 0, 0], Vec( serlaplace( x^2*exp(x)^3-3*x^2*exp(x)^2+3*exp(x)*x^2-x^2))) \\ Joerg Arndt, May 29 2013
(PARI) a(n)={if(n>=2, 3!*n*(n-1)*stirling(n-2, 3, 2), 0)} \\ Andrew Howroyd, Aug 08 2020

Crossrefs

Cf. A052760, A052761.

Sequence in context: A092182 A262387 A133119 * A052765 A265091 A371107

Adjacent sequences: A052774 A052775 A052776 * A052778 A052779 A052780

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Extensions

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

Status

approved