A052776 a(n) = 4!*n*Stirling2(n-1,4).

0, 0, 0, 0, 0, 120, 1440, 10920, 67200, 367416, 1864800, 9003720, 41976000, 190788312, 850930080, 3741028200, 16264684800, 70093951608, 299953709280, 1276255985160, 5404640136000, 22796986414104, 95836788986400, 401733800351400, 1679813323056000

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 733
• Index entries for linear recurrences with constant coefficients, signature (20,-170,800,-2273,3980,-4180,2400,-576).

Crossrefs

Cf. A000919, A052749.

Programs

Maple

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

Mathematica

Join[{0},Table[4!*n*StirlingS2[n-1,4],{n,30}]] (* Harvey P. Dale, Dec 24 2014 *)

PARI

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

Formula

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

Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (1200*n+840*n^2+240*n^3+576+24*n^4)*a(n)+(-1200*n-1300*n^2-450*n^3-50*n^4)*a(n+1)+(35*n^4+420*n+665*n^2+280*n^3)*a(n+2)+(-80*n-10*n^4-140*n^2-70*n^3)*a(n+3)+(n^4+6*n^3+11*n^2+6*n)*a(n+4), a(5)=120}.

From Andrew Howroyd, Aug 08 2020: (Start)

a(n) = n*A000919(n-1) for n > 1.

G.f.: 24*x^5*(5 - 40*x + 105*x^2 - 100*x^3 + 24*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x))^2.

E.g.f.: x*(exp(x) - 1)^4. (End)

Extensions

Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001