OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{j=0..n} (j+1)^(n-1) * Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n.
Ignoring the initial term, equals a diagonal of array A099594, which forms the poly-Bernoulli numbers B(-k,n).
Limit n->infinity a(n)^(1/n)/n^2 = 0.281682... - Vaclav Kotesovec, Dec 30 2012
a(n) = A266695(2*n-1) for n >= 1. - Alois P. Heinz, Apr 17 2024
EXAMPLE
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 46*x^3/3! + 1066*x^4/4! + 41506*x^5/5! +...
where
A(x) = 1 + (1-exp(-2*x))/2 + (1-exp(-3*x))^2/3 + (1-exp(-4*x))^3/4 + (1-exp(-5*x))^4/5 + (1-exp(-6*x))^5/6 +...
MATHEMATICA
Table[Sum[(-1)^(k+n)*(k+1)^(n-1)*k!*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
Table[n!*SeriesCoefficient[Sum[(1-E^(-x*(k+1)))^k/(k+1), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 30 2012 *)
PROG
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-(k+1)*x+x*O(x^n)))^k/(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=sum(j=0, n, (j+1)^(n-1)*sum(i=0, j, (-1)^(n+j-i)*binomial(j, i)*(j-i)^n))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 28 2012
STATUS
approved