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A217924 a(n) = n! * [x^n] exp(2*exp(x) - x - 2). Row sums of triangle A217537.

%I A217924 #63 Jan 09 2025 08:02:35
%S A217924 1,1,3,9,35,153,755,4105,24323,155513,1064851,7760745,59895203,
%T A217924 487397849,4166564147,37298443977,348667014723,3395240969785,
%U A217924 34365336725715,360837080222761,3923531021460707,44108832866004121,511948390801374835,6126363766802713481
%N A217924 a(n) = n! * [x^n] exp(2*exp(x) - x - 2). Row sums of triangle A217537.
%C A217924 The inverse binomial transform of a(n) is A194689.
%C A217924 A087981(n) = Sum_{k=0..n} (-1)^k*s(n+1,k+1)*a(k);
%C A217924 |A000023(n)| = |Sum_{k=0..n} (-1)^(n-k)*s(n,k)*a(k)|
%C A217924 where s(n,k) are the unsigned Stirling numbers of first kind.
%C A217924 a(n) is the number of inequivalent set partitions of {1,2,...,n} where two blocks are considered equivalent when one can be obtained from the other by an alternating (even) permutation. - _Geoffrey Critzer_, Mar 17 2013
%H A217924 Vaclav Kotesovec, <a href="/A217924/b217924.txt">Table of n, a(n) for n = 0..556</a>
%F A217924 G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 06 2013
%F A217924 E.g.f.: exp(2*exp(x) - x - 2). -  _Geoffrey Critzer_, Mar 17 2013
%F A217924 G.f.: 1/Q(0), where Q(k) = 1 - (k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 03 2013
%F A217924 G.f.: T(0)/(1-x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-x-x*k)*(1-2*x-x*k)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 19 2013
%F A217924 a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,k-j)*2^j*(-1)^(k-j)*Stirling2(n-k+j,j). - _Vladimir Kruchinin_, Feb 28 2015
%F A217924 a(n) = exp(-2) * Sum_{k>=0} 2^k * (k - 1)^n / k!. - _Ilya Gutkovskiy_, Jun 27 2020
%F A217924 Conjecture: a(n) = Sum_{k=0..2^n-1} A372205(k). - _Mikhail Kurkov_, Nov 21 2021 [Rewritten by _Peter Luschny_, Apr 22 2024]
%F A217924 a(n) ~ 2 * n^(n-1) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-1)). - _Vaclav Kotesovec_, Jun 26 2022
%e A217924 a(3)=9 because we have: {1,2,3}; {1,3,2}; {1}{2,3}; {1}{3,2}; {2}{1,3}; {2}{3,1}; {3}{1,2}; {3}{2,1}; {1}{2}{3}. - _Geoffrey Critzer_, Mar 17 2013
%p A217924 egf := exp(2*exp(x) - x - 2): ser := series(egf, x, 25):
%p A217924 seq(n!*coeff(ser, x, n), n = 0..23);  # _Peter Luschny_, Apr 22 2024
%t A217924 nn=23;Range[0,nn]!CoefficientList[Series[Exp[2 Exp[x]-x-2],{x,0,nn}],x]  (* _Geoffrey Critzer_, Mar 17 2013 *)
%t A217924 nmax = 25; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-2*k*x^2 , 1 - (k + 1)*x, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 25 2017 *)
%o A217924 (Sage)
%o A217924 def A217924_list(n):
%o A217924     T = A217537_triangle(n)
%o A217924     return [add(T.row(n)) for n in range(n)]
%o A217924 A217924_list(24)
%o A217924 (SageMath)
%o A217924 def A217924_list(prec):
%o A217924     P.<x> = PowerSeriesRing(QQ, prec)
%o A217924     return P( exp(2*exp(x)-x-2) ).egf_to_ogf().list()
%o A217924 print(A217924_list(40)) # _G. C. Greubel_, Jan 09 2025
%o A217924 (Maxima)
%o A217924 a(n):=sum(sum(binomial(n,k-j)*2^j*(-1)^(k-j)*stirling2(n-k+j,j),j,0,k),k,0,n); /* _Vladimir Kruchinin_, Feb 28 2015 */
%o A217924 (Magma)
%o A217924 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A217924 Coefficients(R!(Laplace( Exp(2*Exp(x) -x-2) ))); // _G. C. Greubel_, Jan 09 2025
%Y A217924 Similar recurrences: A124758, A243499, A284005, A329369, A341392, A372205.
%K A217924 nonn
%O A217924 0,3
%A A217924 _Peter Luschny_, Oct 15 2012
%E A217924 Name extended by a formula of _Geoffrey Critzer_ by _Peter Luschny_, Apr 22 2024