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A305923 Expansion of e.g.f. exp(x)/(1 - log(1 + x) - log(1 + x)^2).

%I A305923 #7 Mar 27 2019 03:53:23
%S A305923 1,2,6,21,105,580,4332,33173,333057,3249334,41175698,485901669,
%T A305923 7470988137,102962077608,1870375878472,29342124588357,617978798588225,
%U A305923 10818920340476010,260570216908845406,5009431835664474101,136578252867673635369,2844357524328057280332,87134882338620095240484
%N A305923 Expansion of e.g.f. exp(x)/(1 - log(1 + x) - log(1 + x)^2).
%C A305923 Binomial transform of A005444.
%C A305923 Sequence is signed: first negative term is a(61).
%H A305923 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A305923 a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,k)*Stirling1(k,j)*j!*Fibonacci(j+1).
%F A305923 a(n) ~ (-1)^n * n! * exp(exp(-phi) - phi^2) / (sqrt(5) * (1 - exp(-phi))^(n+1)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Mar 26 2019
%e A305923 E.g.f.: A(x) = 1 + 2*x/1! + 6*x^2/2! + 21*x^3/3! + 105*x^4/4! + 580*x^5/5! + 4332*x^6/6! + ...
%p A305923 a:=series(exp(x)/(1-log(1+x)-log(1+x)^2),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # _Paolo P. Lava_, Mar 26 2019
%t A305923 nmax = 22; CoefficientList[Series[Exp[x]/(1 - Log[1 + x] - Log[1 + x]^2), {x, 0, nmax}], x] Range[0, nmax]!
%t A305923 Table[Sum[Sum[Binomial[n, k] StirlingS1[k, j] j! Fibonacci[j + 1], {j, 0, k}], {k, 0, n}], {n, 0, 22}]
%Y A305923 Cf. A000045, A000556, A005444, A005923, A291981.
%K A305923 sign
%O A305923 0,2
%A A305923 _Ilya Gutkovskiy_, Jun 14 2018