A134055 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

1, 1, 2, 8, 41, 252, 1782, 14121, 123244, 1169832, 11960978, 130742196, 1518514076, 18645970943, 241030821566, 3268214127548, 46338504902485, 685145875623056, 10538790233183702, 168282662416550040, 2784205185437851772, 47646587512911994120

Offset: 0

Author

Paul D. Hanna, Oct 08 2007

Keywords

• nonn

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 252*x^5 + 1782*x^6 + 14121*x^7 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^2*x^2/(1-2*x)^2*exp(-2*x/(1-2*x))/2! + 3^3*x^3/(1-3*x)^3*exp(-3*x/(1-3*x))/3! + 4^4*x^4/(1-4*x)^4*exp(-4*x/(1-4*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(4) = 1*1 + 3*7 + 3*6 + 1*1 = 41;
a(5) = 1*1 + 4*15 + 6*25 + 4*10 + 1*1 = 252;
a(6) = 1*1 + 5*31 + 10*90 + 10*65 + 5*15 + 1*1 = 1782.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..518

Crossrefs

Cf. A037011, A048297, A048993, A122455, A218667, A218670, A245059, A245060.

Programs

Maple

a:= proc(n) option remember; local b; b:=
      proc(h, m) option remember; `if`(h=0,
        binomial(n-1, m-1), m*b(h-1, m)+b(h-1, m+1) )
      end; b(n, 0)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 24 2023

Mathematica

Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[n,k],{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)

PARI

a(n)=if(n==0,1,sum(k=1, n, binomial(n-1, k-1)*polcoeff(1/prod(i=0, k, 1-i*x +x*O(x^(n-k))), n-k)))

PARI

a(n)=polcoeff(sum(k=0,n+1,(k*x)^k/(1-k*x)^k*exp(-k*x/(1-k*x+x*O(x^n)))/k!),n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 04 2012

Formula

O.g.f.: Sum_{n>=0} (n*x)^n/(1-n*x)^n * exp(-n*x/(1-n*x)) / n!. - Paul D. Hanna, Nov 04 2012

From Alois P. Heinz, Jun 24 2023: (Start)

a(n) mod 2 = A037011(n) for n >= 1.

a(n) mod 2 = 1 <=> n in { A048297 } or n = 0. (End)

Extensions

An initial '1' was added and definition changed slightly by Paul D. Hanna, Nov 04 2012