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

1, 2, 6, 21, 105, 580, 4332, 33173, 333057, 3249334, 41175698, 485901669, 7470988137, 102962077608, 1870375878472, 29342124588357, 617978798588225, 10818920340476010, 260570216908845406, 5009431835664474101, 136578252867673635369, 2844357524328057280332, 87134882338620095240484

Offset: 0

Author

Ilya Gutkovskiy, Jun 14 2018

Keywords

• sign

Comments

Binomial transform of A005444.

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

Examples

			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! + ...
		

Links

• N. J. A. Sloane, Transforms

Crossrefs

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

Programs

Maple

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

Mathematica

nmax = 22; CoefficientList[Series[Exp[x]/(1 - Log[1 + x] - Log[1 + x]^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS1[k, j] j! Fibonacci[j + 1], {j, 0, k}], {k, 0, n}], {n, 0, 22}]

Formula

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

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