A052892 E.g.f. is series reversion of log(1+x)*(1-x).

0, 1, 3, 22, 269, 4606, 101407, 2728818, 86783769, 3184595686, 132443395091, 6156200036746, 316272966462565, 17795937622944846, 1088410048965734055, 71893314170319604066, 5100574859506418167601, 386824334429004242804086, 31229208329663841200670619

Offset

0,3

Comments

A simple grammar.

Sum_{k=1..n} a(k)*Sum_{j=k..n} C(k,n-j)*(-1)^(n-j)*Stirling1(j,k)/j! = delta(n,1) for n > 0. - Vladimir Kruchinin, Feb 08 2012

Links

Table of n, a(n) for n=0..18.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 868.

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Formula

E.g.f.: exp(RootOf(-2*_Z+_Z*exp(x*_Z)+1)*x) - 1.

G.f.: x*Sum_{k>0} 1/(2-k*x)^k. - Vladeta Jovovic, Sep 23 2006

a(n) = (-1)^(n-1)*(n-1)!*Sum_{k=0..n-1} Sum_{j=0..k} j!*(Sum_{i=j..n+j-1} Stirling1(i,j)*(-1)^(i)*C(j,n+j-i-1)/i!)*C(k,j)*C(n+k-1,n-1), n > 0. - Vladimir Kruchinin, Feb 08 2012

a(n) = Sum_{i=0..n-1} binomial(n-1,i)*Sum_{k=0..i} Stirling2(i,k)*k!* binomial(n+k-1,k). - Vladimir Kruchinin, Jan 22 2014

a(n) ~ n^(n-1) * (c/2)^(n-1) / (sqrt(c+1) * exp(n) * (c-1)^(2*n-1)), where c = LambertW(2*exp(1)) = 1.3748225281836... - Vaclav Kotesovec, Jan 22 2014

For n >= 1, a(n) = Sum_{k=0..n-1} Pochhammer(n, k)*Stirling2(n, k+1). - Mélika Tebni, Jun 03 2023

Maple

spec := [S, {C=Prod(Z, B), S=Set(C, 1 <= card), B=Sequence(S)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
# second Maple program:
A052892 := proc (n) option remember; `if`(n = 0, 0, add(pochhammer(n, k)*Stirling2(n, k+1), k = 0..n-1)) end:
seq(A052892(n), n = 0..17); # Mélika Tebni, Jun 03 2023

Mathematica

a[n_] := ((-1)^(n-1)*(n-1)!*Sum[ (Sum[ j!*(Sum[ (StirlingS1[i, j]*(-1)^(i)*Binomial[j, n+j-i-1])/i!, {i, j, n+j-1}])*Binomial[k, j], {j, 0, k}])*Binomial[n+k-1, n-1], {k, 0, n-1}]); a[0] = 0; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
CoefficientList[InverseSeries[Series[Log[1+x]*(1-x), {x, 0, 20}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 22 2014 *)

Prog

(Maxima) a(n):=((-1)^(n-1)*(n-1)!*sum((sum(j!*(sum((stirling1(i, j)*(-1)^(i)*binomial(j, n+j-i-1))/i!, i, j, n+j-1))*binomial(k, j), j, 0, k)) *binomial(n+k-1, n-1), k, 0, n-1)); /* Vladimir Kruchinin, Feb 06 2012 */

Crossrefs

Row sums of A198204. - Peter Bala, Aug 01 2012

Cf. A052842.

Sequence in context: A367181 A005264 A195512 * A155806 A196022 A319147

Adjacent sequences: A052889 A052890 A052891 * A052893 A052894 A052895

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Status

approved