OFFSET
1,3
FORMULA
a(n) = n!*T(n,1), T(n,m) = Sum_{k=1..n-m} (T(n-m, k)*m^k/k! - Stirling2(n, k+m-1)*((k+m-1)!/n!)*T(k+m-1, m)), n > m, with T(n,n) = 1.
E.g.f. satisfies A(x) = log(1+x)*exp( A(log(1+x)) ).
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 2*x^3/3! + 4*x^4/4! + 4*x^5/5! + 8*x^6/6! + 202*x^7/7! + ...
A(exp(x)-1) = x + x^2 + x^3 + x^4 + 11/12*x^5 + 3/4*x^6 + 107/180*x^7 + 59/120*x^8+ ...
x*exp(A(x)) = x + x^2 + x^3 + x^4 + 11/12*x^5 + 3/4*x^6 + 107/180*x^7 + 59/120*x^8+ ...
MATHEMATICA
t[n_, m_] := t[n, m] = If[ n == m , 1 , Sum[t[n-m, k]*m^k/k! - StirlingS2[n, k+m-1]*(k+m-1)!/n!*t[k+m-1, m], {k, 1, n-m}]]; a[n_] := n!*t[n, 1]; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Jun 24 2013, after Vladimir Kruchinin *)
PROG
(Maxima) T(n, m):=if n=m then 1 else sum(T(n-m, k)*m^k/k!-stirling2(n, k+m-1)*(k+m-1)!/n!*T(k+m-1, m), k, 1, n-m);
makelist(n!*T(n, 1), n, 1, 15);
(PARI) /* Using A(exp(x)-1) = x*exp(A(x)) */
{a(n)=local(A=x); for(i=1, n, A=log(1+x+x*O(x^n))*exp(subst(A, x, log(1+x+x*O(x^n))))); n!*polcoeff(A, n)} \\ Paul D. Hanna
(PARI) /* Using Vladimir Kruchinin's formula: */
{Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{T(n, k)=if(n<k, 0, if(n==k, 1, sum(j=1, n-k, T(n-k, j)*k^j/j!-Stirling2(n, k+j-1)*(k+j-1)!/n!*T(k+j-1, k)) ))}
{a(n)=n!*T(n, 1)} \\ Paul D. Hanna
CROSSREFS
KEYWORD
sign
AUTHOR
Vladimir Kruchinin, Dec 04 2011
STATUS
approved