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A247075 Expansion of e.g.f.: x^2*G'(x)/G(x)^2, where G(x) satisfies G(x) = x*(1+log(1+G(x))).

%I A247075 #38 Nov 10 2024 16:35:10
%S A247075 1,0,-1,-2,12,96,-220,-7440,-15624,813120,7340112,-104165280,
%T A247075 -2442773520,8815815360,855578733984,4653629425536,-317564443445760,
%U A247075 -5591544140206080,110965435244017920,4730495445765296640,-16883238483957574656
%N A247075 Expansion of e.g.f.: x^2*G'(x)/G(x)^2, where G(x) satisfies G(x) = x*(1+log(1+G(x))).
%H A247075 G. C. Greubel, <a href="/A247075/b247075.txt">Table of n, a(n) for n = 0..425</a>
%F A247075 a(n) = Sum_{k=0..n} k!*binomial(n-1,k)*Stirling1(n,k).
%F A247075 E.g.f.: x^2*G'(x)/G(x)^2 where G(x) = Series_Reversion(x/(1 + log(1+x))); see A177380. - _Paul D. Hanna_, Nov 17 2014
%p A247075 A:= n -> add(k!*binomial(n-1,k)*combinat:-stirling1(n,k),k=0..n):
%p A247075 seq(A(n),n=0..30); # _Robert Israel_, Nov 17 2014
%t A247075 Table[Sum[StirlingS1[n, k] k! Binomial[n-1, k], {k, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, Nov 17 2014 *)
%o A247075 (Maxima)
%o A247075 a(n):=sum(k!*binomial(n-1,k)*stirling1(n,k),k,0,n);
%o A247075 (Magma) [(&+[Factorial(j)*Binomial(n-1,j)*StirlingFirst(n,j): j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Mar 08 2023
%o A247075 (SageMath)
%o A247075 def A247075(n): return sum( (-1)^(n-k)*factorial(k)*binomial(n-1,k)*stirling_number1(n,k) for k in range(n+1))
%o A247075 [A247075(n) for n in range(21)] # _G. C. Greubel_, Mar 08 2023
%Y A247075 Cf. A177380.
%K A247075 sign
%O A247075 0,4
%A A247075 _Vladimir Kruchinin_, Nov 17 2014