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

%I A198860 #37 Jan 12 2025 19:11:15
%S A198860 1,1,3,17,144,1634,23312,401274,8096680,187472136,4900535832,
%T A198860 142766286552,4587190461840,161161214978880,6146415080939520,
%U A198860 252902928346825104,11167368115492742400,526752556713346955520,26433830208985721222400,1406218428780691953635712
%N A198860 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x))).
%C A198860 Compare to e.g.f. G(x) of A052802, which satisfies: G(x) = 1/(1 + log(1 - x*G(x))).
%F A198860 E.g.f. satisfies: A(x*(1 - log(1+x))) = 1/(1 - log(1+x)).
%F A198860 E.g.f.: A(x) = (1/x)*Series_Reversion[x - x*log(1+x)].
%F A198860 a(n) = n!*[x^n] 1/(1 - log(1+x))^(n+1)/(n+1).
%F A198860 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
%F A198860 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
%F A198860 a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * Stirling1(n,k). - _Seiichi Manyama_, Nov 06 2023
%e A198860 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 144*x^4/4! + 1634*x^5/5! + ...
%e A198860 where log(1 + x*A(x)) equals
%e A198860 1 - 1/A(x) = x + x^2/2! + 5*x^3/3! + 38*x^4/4! + 404*x^5/5! + 5514*x^6/6! + ...
%t A198860 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_ *)
%t A198860 CoefficientList[1/x*InverseSeries[Series[x-x*Log[1+x], {x, 0, 20}], x],x] * Range[0, 19]! (* _Vaclav Kotesovec_, Dec 28 2013 *)
%o A198860 (PARI) {a(n)=n!*polcoeff((1/x)*serreverse(x-x*log(1+x +x*O(x^n))),n)}
%o A198860 (PARI) {a(n)=n!*polcoeff(1/(1-log(1+x +x*O(x^n)))^(n+1)/(n+1),n)}
%o A198860 (PARI) /* by _Vladimir Kruchinin_'s formula: */
%o A198860 {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)*stirling(n,i,1))))/(n+1)}
%o A198860 (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 */
%Y A198860 Cf. A030178, A052802.
%K A198860 nonn
%O A198860 0,3
%A A198860 _Paul D. Hanna_, Oct 30 2011