OFFSET
0,5
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)).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
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.
EXAMPLE
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.
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 04 2012
STATUS
approved