A198860 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x))).

1, 1, 3, 17, 144, 1634, 23312, 401274, 8096680, 187472136, 4900535832, 142766286552, 4587190461840, 161161214978880, 6146415080939520, 252902928346825104, 11167368115492742400, 526752556713346955520, 26433830208985721222400, 1406218428780691953635712

Offset

0,3

Comments

Compare to e.g.f. G(x) of A052802, which satisfies: G(x) = 1/(1 + log(1 - x*G(x))).

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f. satisfies: A(x*(1 - log(1+x))) = 1/(1 - log(1+x)).

E.g.f.: A(x) = (1/x)*Series_Reversion[x - x*log(1+x)].

a(n) = n!*[x^n] 1/(1 - log(1+x))^(n+1)/(n+1).

a(n) = Sum_{k=0..n} (binomial(n+k,n) * Sum_{j=0..k} (-1)^(j)*binomial(k,j) * (Sum_{i=0..j} (-1)^i*i!*binomial(j,i)*Stirling1(n,i)))/(n+1). - Vladimir Kruchinin, Feb 04 2012

a(n) ~ n^(n-1) / ((1-c)*sqrt(1+c) * exp(n) * (1/c+c-2)^n), where c = LambertW(1). - Vaclav Kotesovec, Dec 28 2013

a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * Stirling1(n,k). - Seiichi Manyama, Nov 06 2023

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 144*x^4/4! + 1634*x^5/5! + ...
where log(1 + x*A(x)) equals
1 - 1/A(x) = x + x^2/2! + 5*x^3/3! + 38*x^4/4! + 404*x^5/5! + 5514*x^6/6! + ...

Mathematica

a[n_] := Sum[ Binomial[n+k, n]*Sum[ (-1)^(j)*Binomial[k, j]*Sum[ (-1)^i*i!*Binomial[j, i]*StirlingS1[n, i], {i, 0, j}], {j, 0, k}], {k, 0, n}]/(n+1); Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jun 24 2013, after Vladimir Kruchinin *)
CoefficientList[1/x*InverseSeries[Series[x-x*Log[1+x], {x, 0, 20}], x], x] * Range[0, 19]! (* Vaclav Kotesovec, Dec 28 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff((1/x)*serreverse(x-x*log(1+x +x*O(x^n))), n)}
(PARI) {a(n)=n!*polcoeff(1/(1-log(1+x +x*O(x^n)))^(n+1)/(n+1), n)}
(PARI) /* by Vladimir Kruchinin's formula: */
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=sum(k=0, n, binomial(n+k, n)*sum(j=0, k, (-1)^(j)*binomial(k, j)*sum(i=0, j, (-1)^i*i!*binomial(j, i)*Stirling1(n, i))))/(n+1)} \\ Paul D. Hanna
(Maxima) a(n):=sum(binomial(n+k, n)*sum((-1)^(j)*binomial(k, j)*sum((-1)^i*i!*binomial(j, i)* stirling1(n, i), i, 0, j), j, 0, k), k, 0, n))/(n+1); // Vladimir Kruchinin, Feb 04 2012

Crossrefs

Cf. A030178, A052802.

Sequence in context: A361767 A268254 A333331 * A298691 A362282 A051442

Adjacent sequences: A198857 A198858 A198859 * A198861 A198862 A198863

Keyword

nonn

Author

Paul D. Hanna, Oct 30 2011

Status

approved