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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. 13
1, 1, 2, 8, 41, 252, 1782, 14121, 123244, 1169832, 11960978, 130742196, 1518514076, 18645970943, 241030821566, 3268214127548, 46338504902485, 685145875623056, 10538790233183702, 168282662416550040, 2784205185437851772, 47646587512911994120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..21.

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

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.

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. 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

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Last modified July 24 14:43 EDT 2021. Contains 346273 sequences. (Running on oeis4.)