A201827 E.g.f. satisfies: A(log(1+x)) = x*exp(A(x)).

1, 3, 19, 203, 3116, 63400, 1626815, 50869481, 1890809326, 81984117362, 4085969612187, 231307144828171, 14729010409178287, 1046405525995004427, 82366263901831053010, 7140311864811600275193, 678154920246172000643119, 70238663891082116737406875, 7900753256152524322019355019

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Formula

a(n) = n!*T(n,1), T(n,m) = Sum_{k=1..n-m} (T(n-m,k)*m^k/k! - Stirling1(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) = (exp(x)-1)*exp( A(exp(x)-1) ).

Example

E.g.f.: A(x) = x + 3*x^2/2! + 19*x^3/3! + 203*x^4/4! + 3116*x^5/5! + ...
A(log(1+x)) = x + 2*x^2/2! + 12*x^3/3! + 116*x^4/4! + 1625*x^5/5! + ...

Mathematica

t[n_, k_] /; n < k = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = Sum[ t[n-k, j]*k^j/j! - StirlingS1[n, k+j-1]*(k+j-1)!/n!*t[k+j-1, k], {j, 1, n-k}]; a[n_] := n!*t[n, 1]; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Dec 21 2011, after Vladimir Kruchinin's formula *)

Prog

(Maxima)
array(B, 25, 25);
fillarray (B, makelist (-1, i, 1, 1000));
T(n, m):=if B[n, m]=-1 then B[n, m]:(if n=m then 1 else sum(T(n-m, k)*m^k/k!-stirling1(n, k+m-1)*(k+m-1)!/n!*T(k+m-1, m), k, 1, n-m)) else B[n, m];
makelist(n!*T(n, 1), n, 1, 21);
(PARI) /* Using A(x) = (exp(x)-1)*exp(A(exp(x)-1)) [from Paul D. Hanna]: */
{a(n)=local(A=x, X=x+x*O(x^n)); for(i=1, n, A=(exp(X)-1)*exp(subst(A, x, exp(X)-1))); n!*polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
(PARI) /* Using Vladimir Kruchinin's formula [program by Paul D. Hanna]: */
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{T(n, k)=if(n<k, 0, if(n==k, 1, sum(j=1, n-k, T(n-k, j)*k^j/j! - Stirling1(n, k+j-1)*(k+j-1)!/n!*T(k+j-1, k))))}
{a(n) = n!*T(n, 1)}

Crossrefs

Sequence in context: A208897 A027546 A301419 * A230321 A108993 A245308

Adjacent sequences: A201824 A201825 A201826 * A201828 A201829 A201830

Keyword

nonn

Author

Vladimir Kruchinin, Dec 05 2011

Status

approved