A052760 -id:A052760

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A052777

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

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.

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

A052792

Expansion of e.g.f.: x^2*(exp(x)-1)^4.

+10
1

0, 0, 0, 0, 0, 0, 720, 10080, 87360, 604800, 3674160, 20512800, 108044640, 545688000, 2671036368, 12763951200, 59856451200, 276499641600, 1261691128944, 5699120476320, 25525119703200, 113497442856000, 501533701110288, 2204246146687200, 9641611208433600

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

Original name: a simple grammar.

LINKS

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 749

Index entries for linear recurrences with constant coefficients, signature (30,-405,3250,-17247,63690,-167615,316350,-424428,394280,-240480,86400,-13824).

FORMULA

E.g.f.: x^2*exp(x)^4-4*x^2*exp(x)^3+6*x^2*exp(x)^2-4*exp(x)*x^2+x^2.

Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, (1200*n+840*n^2+240*n^3+576+24*n^4)*a(n)+(1200-50*n^4+100*n-850*n^2-400*n^3)*a(n+1)+(210*n^3+175*n^2+35*n^4-420*n)*a(n+2)+(10*n^2-40*n^3+40*n-10*n^4)*a(n+3)+(-n^2+n^4-2*n+2*n^3)*a(n+4)}.

a(n) = n*A052776(n-1) = 4!*n*(n-1)*Stirling2(n-2,4) for n >= 2. - Andrew Howroyd, Aug 08 2020

MAPLE

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