cp's OEIS Frontend

A232549 E.g.f.: Sum_{n>=0} ( -log(1 - n*x)/n )^n / n!.

%I A232549 #11 Dec 25 2023 17:44:23
%S A232549 1,1,2,9,69,780,12015,240254,6022121,184115728,6718994385,
%T A232549 287695076790,14253791802085,807709767584928,51832973461142601,
%U A232549 3734821283563097350,299928689437970776785,26669987585007924913536,2610872906937221730434977,279949862628226582759321542,32727135115533487802060517885
%N A232549 E.g.f.: Sum_{n>=0} ( -log(1 - n*x)/n )^n / n!.
%C A232549 Compare e.g.f. to the identity:
%C A232549 if G(x) = Sum_{n>=0} Integral( G(x)^t dx )^n/n!, then G(x)^t = 1/(1 - t*x).
%H A232549 Vincenzo Librandi, <a href="/A232549/b232549.txt">Table of n, a(n) for n = 0..200</a>
%F A232549 E.g.f.: Sum_{n>=0} Integral( 1/(1 - n*x) dx )^n/n!, where the constant of integration is zero.
%F A232549 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(n-k) * Stirling1(n,k).
%e A232549 G.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 69*x^4/4! + 780*x^5/5! + 12015*x^6/6! +...
%e A232549 where
%e A232549 A(x) = 1 - log(1-x) + log(1-2*x)^2/(2!*2^2) - log(1-3*x)^3/(3!*3^3) + log(1-4*x)^4/(4!*4^4) - log(1-5*x)^5/(5!*5^5) + log(1-6*x)^6/(6!*6^6) +...
%t A232549 Flatten[{1,Table[Sum[(-1)^(n-k) * k^(n-k) * StirlingS1[n,k],{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 23 2014 *)
%o A232549 (PARI) {a(n)=local(A=1+x);A=sum(m=0,n,intformal( 1/(1-m*x + x*O(x^n)))^m/m!);n!*polcoeff(A,n)}
%o A232549 for(n=0,20,print1(a(n),", "))
%o A232549 (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A232549 {a(n)=sum(k=0, n, (-1)^(n-k) * k^(n-k) * Stirling1(n, k))}
%o A232549 for(n=0,20,print1(a(n),", "))
%Y A232549 Cf. A128943.
%K A232549 nonn
%O A232549 0,3
%A A232549 _Paul D. Hanna_, Nov 25 2013