A052785 a(n) = 5!*n*Stirling2(n-1, 5).

0, 0, 0, 0, 0, 0, 720, 12600, 134400, 1134000, 8341200, 56133000, 355291200, 2151864000, 12614281680, 72135063000, 404672486400, 2236228722000, 12209943566160, 66024457842600, 354214283304000, 1887999348060000, 10008933180578640, 52820388477271800, 277680637970208000

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..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 742
• Index entries for linear recurrences with constant coefficients, signature (30,-395,3000,-14523,46710,-100805,143700,-129076,65760,-14400).

Crossrefs

Cf. A001118, A052749.

Programs

Maple

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

PARI

a(n)={if(n>=1, 5!*n*stirling(n-1, 5, 2), 0)} \\ Andrew Howroyd, Aug 08 2020

Formula

E.g.f.: x*exp(x)^5-5*x*exp(x)^4+10*exp(x)^3*x-10*exp(x)^2*x+5*x*exp(x)-x.

Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, ( - 14400 - 27000*n^2 - 32880*n - 120*n^5 - 1800*n^4 - 10200*n^3)*a(n) + (42196*n^2 + 19454*n^3 + 32880*n + 274*n^5 + 3836*n^4)*a(n + 1) + ( - 13500*n - 13275*n^3 - 24075*n^2 - 225*n^5 - 2925*n^4)*a(n + 2) + (85*n^5 + 3400*n + 1020*n^4 + 4165*n^3 + 6630*n^2)*a(n + 3) + ( - 915*n^2 - 450*n - 615*n^3 - 15*n^5 - 165*n^4)*a(n + 4) + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a(n + 5)}.

From Andrew Howroyd, Aug 08 2020: (Start)

a(n) = n*A001118(n-1) for n > 1.

G.f.: 120*x^6*(2 - 5*x)*(3 - 30*x + 95*x^2 - 100*x^3 + 24*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x))^2.

E.g.f.: x*(exp(x) - 1)^5. (End)

Extensions

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