OFFSET
0,3
COMMENTS
Compare the g.f. to: Sum_{n>=0} exp(-(1+n*x)) * (1+n*x)^n / n! = 1/(1-x).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..150
FORMULA
a(n) = Sum_{k=1..n} Stirling2(n, k) * C(n+k-1, k-1) for n>0 with a(0)=1.
Row sums of Triangle A245111.
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 161*x^4 + 1302*x^5 + 11810*x^6 +...
where
A(x) = exp(-1) + exp(-1/(1-x))/(1-x) + (exp(-1/(1-2*x))/(1-2*x)^2)/2!
+ (exp(-1/(1-3*x))/(1-3*x)^3)/3! + (exp(-1/(1-4*x))/(1-4*x)^4)/4!
+ (exp(-1/(1-5*x))/(1-5*x)^5)/5! + (exp(-1/(1-6*x))/(1-6*x)^6)/6!
+ (exp(-1/(1-7*x))/(1-7*x)^7)/7! + (exp(-1/(1-8*x))/(1-8*x)^8)/8! +...
simplifies to a power series in x with integer coefficients.
PROG
(PARI) /* From definition (requires setting suitable precision) */ \p100
{a(n)=local(A=1+x, X=x+x*O(x^n)); A=suminf(k=0, exp(-1/(1-k*X))/(1-k*X)^k/k!); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From a(n) = Sum_{k=1..n} Stirling2(n, k) * C(n+k-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) * binomial(n+k-1, k-1)))}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* As row sums of triangle A245111: */
{A245111(n, k)=local(A=1+x*y); A=sum(k=0, n, 1/(1-k*x+x*O(x^n))^k*y^k/k!*exp(-y/(1-k*x+x*O(x^n))+y*O(y^n))); polcoeff(polcoeff(A, n, x), k, y)}
{a(n) = sum(k=0, n, A245111(n, k))}
/* Print Initial Rows of Triangle A245111: */
{for(n=0, 10, for(k=0, n, print1(A245111(n, k), ", ")); print(""))}
/* Row Sums yield A245110: */
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 12 2014
STATUS
approved