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%I #25 Apr 17 2024 20:17:07
%S 1,1,4,46,1066,41506,2441314,202229266,22447207906,3216941445106,
%T 578333776748674,127464417117501586,33800841048945424546,
%U 10617398393395844992306,3898852051843774954576834,1654948033478889053351543506,804119629083230641164978005986
%N E.g.f.: Sum_{n>=0} (1 - exp(-(n+1)*x))^n/(n+1).
%H Vincenzo Librandi, <a href="/A188634/b188634.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{j=0..n} (j+1)^(n-1) * Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n.
%F Ignoring the initial term, equals a diagonal of array A099594, which forms the poly-Bernoulli numbers B(-k,n).
%F Limit n->infinity a(n)^(1/n)/n^2 = 0.281682... - _Vaclav Kotesovec_, Dec 30 2012
%F a(n) = A266695(2*n-1) for n >= 1. - _Alois P. Heinz_, Apr 17 2024
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 46*x^3/3! + 1066*x^4/4! + 41506*x^5/5! +...
%e where
%e 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 +...
%t Table[Sum[(-1)^(k+n)*(k+1)^(n-1)*k!*StirlingS2[n, k],{k,0,n}],{n,0,20}]
%t 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 *)
%o (PARI) {a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-(k+1)*x+x*O(x^n)))^k/(k+1)), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (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))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A099594, A092552, A191908, A266695.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 28 2012
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