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

0,3

Links

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

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)

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A067119 A093935 A099240 * A136281 A125698 A231495

Adjacent sequences: A134052 A134053 A134054 * A134056 A134057 A134058

Keyword

nonn

Author

Paul D. Hanna, Oct 08 2007

Extensions

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

Status

approved