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%I #12 Jan 27 2020 07:11:54
%S 1,1,2,5,15,54,220,973,4607,23230,124088,698471,4124961,25474314,
%T 164063103,1099233251,7645091839,55085061358,410472347944,
%U 3158307976315,25057152530411,204717532709542,1720324316575275,14853374782672785,131632834029683663,1196258970969508760
%N O.g.f.: 1/(1-x) * Sum_{n>=0} 1/(1-(n+1)*x)^n * x^n/n! * exp(-x/(1-(n+1)*x)).
%C Compare g.f. to the curious identity:
%C 1/(1+x^2) = Sum_{n>=0} (1-(n+1)*x)^n * x^n/n! * exp(-x*(1-(n+1)*x)).
%F Binomial transform of A218667.
%e O.g.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 54*x^5 + 220*x^6 +...
%e where the o.g.f. equals the series:
%e A(x)*(1-x) = exp(-x/(1-x)) + x/(1-2*x)*exp(-x/(1-2*x)) + x^2/(1-3*x)^2/2!*exp(-x/(1-3*x)) + x^3/(1-4*x)^3/3!*exp(-x/(1-4*x)) + x^4/(1-5*x)^4/4!*exp(-x/(1-5*x)) + x^5/(1-6*x)^5/5!*exp(-x/(1-6*x)) + x^6/(1-7*x)^6/6!*exp(-x/(1-7*x)) +...
%e which simplifies to a power series in x with integer coefficients.
%t m = 26; Sum[1/(1 - (n+1)x)^n x^n Exp[-x/(1 - (n+1)x)]/n!, {n, 0, m}]/(1-x) + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Jan 27 2020 *)
%o (PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); A=1/(1-x)*sum(k=0, n, 1/(1-(k+1)*X)^k*x^k/k!*exp(-X/(1-(k+1)*X))); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A218667.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 25 2012
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