A188634 E.g.f.: Sum_{n>=0} (1 - exp(-(n+1)*x))^n/(n+1).
1, 1, 4, 46, 1066, 41506, 2441314, 202229266, 22447207906, 3216941445106, 578333776748674, 127464417117501586, 33800841048945424546, 10617398393395844992306, 3898852051843774954576834, 1654948033478889053351543506, 804119629083230641164978005986
Offset: 0
Keywords
Examples
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 +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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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 *) -
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), ", "))
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