A075907 Fourth column of triangle A075499.

1, 40, 1040, 22400, 435456, 7956480, 139694080, 2387968000, 40075329536, 663887544320, 10896534405120, 177653730508800, 2882307270639616, 46596186764738560, 751299029274460160, 12089975328525516800

Offset: 0

Author

Wolfdieter Lang, Oct 02 2002

Keywords

• nonn
• easy

Comments

The e.g.f. given below is Sum_{m=0..3} A075513(4,m)*exp(4*(m+1)*x)/3!.

Links

• Index entries for linear recurrences with constant coefficients, signature (40, -560, 3200, -6144).

Crossrefs

Cf. A019677, A075908.

Programs

Mathematica

Table[(-4^n+24*8^n-81*12^n+64*16^n)/6,{n,0,20}] (* or *) LinearRecurrence[ {40,-560,3200,-6144},{1,40,1040,22400},20] (* Harvey P. Dale, Jun 04 2013 *)

Formula

a(n) = A075499(n+4, 4) = (4^n)*S2(n+4, 4) with S2(n, m) := A008277(n, m) (Stirling2).

a(n) = (-4^n + 24*8^n - 81*12^n + 64*16^n)/3!.

G.f.: 1/Product_{k=1..4} (1 - 4*k*x).

E.g.f.: (d^4/dx^4)(((exp(4*x)-1)/4)^4)/4! = (-exp(4*x) + 24*exp(8*x) - 81*exp(12*x) + 64*exp(16*x))/3!.

a(0)=1, a(1)=40, a(2)=1040, a(3)=22400, a(n) = 40*a(n-1) - 560*a(n-2) + 3200*a(n-3) - 6144*a(n-4). - Harvey P. Dale, Jun 04 2013