A185040 O.g.f.: 1/(1-x) * Sum_{n>=0} 1/(1-(n+1)*x)^n * x^n/n! * exp(-x/(1-(n+1)*x)).

1, 1, 2, 5, 15, 54, 220, 973, 4607, 23230, 124088, 698471, 4124961, 25474314, 164063103, 1099233251, 7645091839, 55085061358, 410472347944, 3158307976315, 25057152530411, 204717532709542, 1720324316575275, 14853374782672785, 131632834029683663, 1196258970969508760

Offset: 0

Author

Paul D. Hanna, Dec 25 2012

Keywords

• nonn

Comments

Compare g.f. to the curious identity:

1/(1+x^2) = Sum_{n>=0} (1-(n+1)*x)^n * x^n/n! * exp(-x*(1-(n+1)*x)).

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 54*x^5 + 220*x^6 +...
where the o.g.f. equals the series:
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)) +...
which simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218667.

Programs

Mathematica

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 *)

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)}
for(n=0, 30, print1(a(n), ", "))

Formula

Binomial transform of A218667.