%I #34 Sep 18 2024 10:47:01
%S 1,1,2,4,9,22,57,157,453,1368,4296,13995,47138,163779,585741,2152349,
%T 8113188,31326760,123748871,499539900,2058542819,8651755865,
%U 37054078481,161591063250,717032333816,3235298221401,14834735654080,69085973044125
%N O.g.f.: A(x) = Sum_{n>=0} x^n*Product_{k=0..n} (1 + k*x).
%C The Kn11 triangle sums of A094638 are given by the terms of this sequence. For the definitions of this and other triangle sums see A180662. [_Johannes W. Meijer_, Apr 20 2011]
%H Vaclav Kotesovec, <a href="/A124380/b124380.txt">Table of n, a(n) for n = 0..868</a>
%H Giulio Cerbai, Anders Claesson, and Bruce E. Sagan, <a href="https://arxiv.org/abs/2408.06959">Self-modified difference ascent sequences</a>, arXiv:2408.06959 [math.CO], 2024. See p. 15.
%F O.g.f.: A(x) = 1 + x*(1+x)/(G(0) - x*(1+x)) ; G(k) = 1+x*(k*x+x+1) - x*(k*x + 2*x + 1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 02 2011
%F G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (1+x*k)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2013
%F G.f.: 1/(x*Q(0)-1)/x^4 + (1+x-x^3)/x^4, where Q(k)= 1 - x/(1 - (k+1)*x - x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 19 2013
%F Conjecture: log(a(n)) ~ n*log(n)/2 - n*(1 + log(2))/2. - _Vaclav Kotesovec_, Sep 18 2024
%e A(x) = 1 + x*(1+x) + x^2*(1+x)*(1+2x) + x^3*(1+x)*(1+2x)*(1+3x) +...
%t nmax = 30; CoefficientList[Series[Sum[x^(2*k)*Pochhammer[1 + 1/x, k], {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 14 2024 *)
%t Table[Sum[(-1)^k * StirlingS1[n+1-k, n+1-2*k], {k, 0, (n+1)/2}], {n, 0, 30}] (* _Vaclav Kotesovec_, Sep 18 2024 *)
%o (PARI) a(n)=polcoeff(sum(k=0,n,x^k*prod(j=0,k,1+j*x+x*O(x^n))),n)
%Y Cf. A024427, A171367.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 28 2006