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Exponential generating function is -f(x) * Integral_{t = 0..x} exp(exp(-t) - 1) dt, where f(x) = exp(1 - x - exp(-x)) is the exponential generating function for A014182.
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%I #51 May 27 2024 23:00:02

%S -1,1,1,-5,5,21,-105,141,777,-5513,13209,39821,-527525,2257425,-41511,

%T -70561285,531862173,-1559180499,-8858267353,147780183829,

%U -936560917615,1352130196615,38710924110081,-487251979381019,2846575686392251,872653153712201

%N Exponential generating function is -f(x) * Integral_{t = 0..x} exp(exp(-t) - 1) dt, where f(x) = exp(1 - x - exp(-x)) is the exponential generating function for A014182.

%H Branko Dragovich, <a href="https://arxiv.org/abs/1702.02569">On Summation of p-Adic Series</a>, arXiv:1702.02569 [math.NT], 2017.

%H Branko Dragovich, Andrei Yu. Khrennikov, and Natasa Z. Misic, <a href="http://arxiv.org/abs/1508.05079">Summation of p-Adic Functional Series in Integer Points</a>, arXiv:1508.05079 [math.NT], 2015.

%H B. Dragovich and N. Z. Misic, <a href="http://dx.doi.org/10.1134/S2070046614040025">p-Adic invariant summation of some p-adic functional series</a>, P-Adic Numbers, Ultrametric Analysis, and Applications, October 2014, Volume 6, Issue 4, pp 275-283.

%F E.g.f. A(x) = y satisfies y'' + y'(2-exp(-x)) + y = 0. - _Michael Somos_, Mar 11 2004

%F a(n) = Sum_{k = 0..n} (-1)^(n-k+1)*Stirling2(n+1, k+1)*A003422(k). - _Vladeta Jovovic_, Jan 06 2005

%F The sequence b(n) = (-1)^n*a(n) satisfies the recurrence: b(n) = -Sum_{i = 1..n} b(i-1)*C(n, i), b(0) = -1. - _Ralf Stephan_, Feb 24 2005

%F From _Peter Bala_, Mar 23 2024: (Start)

%F It appears that a(n) = Sum_{k = 1..n+1} binomial(n+1, k)*a(k). See Dragovich 2017, Table 1.

%F If true then the following hold: setting a(0) = -1 then

%F a(n) = Sum_{k = 1..n-1} (-1)^(n-k)*binomial(n-1, k-1)*a(k-1);

%F the o.g.f F(x) = - ( x/(1 + x)^2 + x^2/((1 + x)*(1 + 2*x)^2) + x^3/((1 + x)*(1 + 2*x)*(1 + 3*x)^2) + ... ) - Cf. A040027;

%F F(x) = - x/(1 + x)^2 + x/(1 + x)^2*F(x/(1 + x)). (End)

%t a[n_] := Sum[(-1)^(n - k + 1) * StirlingS2[n + 1, k + 1] * ((-1)^k * k! * Subfactorial[-k - 1] - Subfactorial[-1]), {k, 0, n}]; Table[a[n] // FullSimplify, {n, 1, 26}] (* _Jean-François Alcover_, Jan 09 2014, after _Vladeta Jovovic_ *)

%t nmax = 25; Rest[CoefficientList[Series[E^(-E^(-x) - x) * (Gamma[0, -1] - Gamma[0, -E^(-x)]), {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, May 10 2024 *)

%o (PARI) a(n)=local(A,B);if(n<0,0, A=exp(-x+x*O(x^n)); B=exp(A-1);n!*polcoeff(-intformal(B)*A/B,n))

%Y Cf. A000670, A040027, A163940.

%K sign

%O 1,4

%A _Noam D. Elkies_

%E More terms from _Jason Earls_, Jun 28 2001