A218667 O.g.f.: Sum_{n>=0} 1/(1-n*x)^n * x^n/n! * exp(-x/(1-n*x)).

1, 0, 1, 1, 4, 13, 46, 181, 778, 3585, 17566, 91171, 499324, 2873839, 17317743, 108933098, 713481122, 4855161425, 34257461754, 250177938679, 1887886966690, 14699340919293, 117933068390123, 973776266303732, 8265721830953558, 72052688932613079, 644393453082317301

Offset: 0

Author

Paul D. Hanna, Nov 04 2012

Keywords

• nonn

Comments

Compare g.f. to the curious identity:

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

Examples

			O.g.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 13*x^5 + 46*x^6 + 181*x^7 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-2*x)^2/2!*exp(-x/(1-2*x)) + x^3/(1-3*x)^3/3!*exp(-x/(1-3*x)) + x^4/(1-4*x)^4/4!*exp(-x/(1-4*x)) + x^5/(1-5*x)^5/5!*exp(-x/(1-5*x)) + x^6/(1-6*x)^6/6!*exp(-x/(1-6*x)) +...
simplifies to a power series in x with integer coefficients.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A245111, A245110, A218668, A218669, A218670, A185040, A217900.

Programs

PARI

{a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k*X)^k*x^k/k!*exp(-X/(1-k*X)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

PARI

/* From a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(-1, k-1) */
{Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
{a(n)=if(n==0, 1, sum(k=1, n, Stirling2(n-k, k) * binomial(n-1, k-1)))}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(n-1, k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jul 30 2014

Antidiagonal sums of Triangle A245111.