A052892 - OEIS

%I #54 Jun 03 2023 18:02:36

%S 0,1,3,22,269,4606,101407,2728818,86783769,3184595686,132443395091,

%T 6156200036746,316272966462565,17795937622944846,1088410048965734055,

%U 71893314170319604066,5100574859506418167601,386824334429004242804086,31229208329663841200670619

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

%C A simple grammar.

%C 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

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=868">Encyclopedia of Combinatorial Structures 868</a>.

%H Vladimir Kruchinin, <a href="https://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.

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

%F G.f.: x*Sum_{k>0} 1/(2-k*x)^k. - _Vladeta Jovovic_, Sep 23 2006

%F 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

%F 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

%F 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

%F For n >= 1, a(n) = Sum_{k=0..n-1} Pochhammer(n, k)*Stirling2(n, k+1). - _Mélika Tebni_, Jun 03 2023

%p spec := [S,{C=Prod(Z,B),S=Set(C,1 <= card),B=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%p # second Maple program:

%p A052892 := proc (n) option remember; `if`(n = 0, 0, add(pochhammer(n, k)*Stirling2(n, k+1), k = 0..n-1)) end:

%p seq(A052892(n), n = 0..17); # _Mélika Tebni_, Jun 03 2023

%t 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_ *)

%t CoefficientList[InverseSeries[Series[Log[1+x]*(1-x), {x, 0, 20}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 22 2014 *)

%o (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 */

%Y Row sums of A198204. - _Peter Bala_, Aug 01 2012

%Y Cf. A052842.

%K easy,nonn

%O 0,3

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