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

%I #7 Mar 27 2019 03:53:23

%S 1,2,6,21,105,580,4332,33173,333057,3249334,41175698,485901669,

%T 7470988137,102962077608,1870375878472,29342124588357,617978798588225,

%U 10818920340476010,260570216908845406,5009431835664474101,136578252867673635369,2844357524328057280332,87134882338620095240484

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

%C Binomial transform of A005444.

%C Sequence is signed: first negative term is a(61).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,k)*Stirling1(k,j)*j!*Fibonacci(j+1).

%F 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 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 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 nmax = 22; CoefficientList[Series[Exp[x]/(1 - Log[1 + x] - Log[1 + x]^2), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Sum[Binomial[n, k] StirlingS1[k, j] j! Fibonacci[j + 1], {j, 0, k}], {k, 0, n}], {n, 0, 22}]

%Y Cf. A000045, A000556, A005444, A005923, A291981.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Jun 14 2018