A052792 Expansion of e.g.f.: x^2*(exp(x)-1)^4.

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

Offset: 0

Author

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

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A052760, A052776.

Programs

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);

PARI

a(n)={if(n>=2, 4!*n*(n-1)*stirling(n-2,4,2), 0)} \\ Andrew Howroyd, Aug 08 2020

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

Extensions

Name changed and terms a(21) and beyond from Andrew Howroyd, Aug 08 2020