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A088354 G.f. = continued fraction: A(x)=1/(1-x-x/(1-x^2-x^2/(1-x^3-x^3/(1-x^4-x^4/(...))))).

%I A088354 #25 Jul 31 2019 10:56:24
%S A088354 1,2,4,10,24,60,150,376,944,2372,5962,14988,37684,94752,238252,599090,
%T A088354 1506440,3788036,9525280,23952020,60229184,151450970,380835368,
%U A088354 957640640,2408063340,6055266600,15226449480,38288118984,96278523274,242100012876,608779761460,1530825191912
%N A088354 G.f. = continued fraction: A(x)=1/(1-x-x/(1-x^2-x^2/(1-x^3-x^3/(1-x^4-x^4/(...))))).
%C A088354 From _Peter Bala_, Jul 29 2019: (Start)
%C A088354 a(n) is the number of triangle stacks of large Schröder type containing n down-triangles. See Links for a definition and an illustration.
%C A088354 Cf. A088352 for triangle stacks of large Schröder type on n triangles. Cf. A224704, which enumerates triangle stacks (of small Schröder type) on n triangles. (End)
%H A088354 P. Bala, <a href="/A088354/a088354.pdf">Illustration for a(3) = 10.</a>
%H A088354 P. Bala, <a href="/A326676/a326676.pdf">Triangle Stacks of large Schröder type</a>
%F A088354 G.f.: 1/Q(0), where Q(k)= 1 - x^(k+1) - x^(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Apr 30 2013
%F A088354 G.f.: T(0)/(1-x), where T(k) = 1 - x^(k+1)/(x^(k+1) - (1-x^(k+1))*(1-x^(k+2))/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 14 2013
%F A088354 a(n) ~ c * d^n, where d = 2.514579643878729188510437194343099820141030855900783271935495710723840992... and c = 0.589519721244409964128200577034763735132770782513329859477444288778116... - _Vaclav Kotesovec_, Jul 01 2019
%F A088354 From _Peter Bala_, Jul 29 2019: (Start)
%F A088354 O.g.f. as a continued fraction:
%F A088354 1/(1 - 2*d/(1 - d^2/(1 - (d^2 + d^3)/(1 - d^4/(1 - (d^3 + d^5)/(1 - d^6/( (...) ))))))).
%F A088354 O.g.f. as a ratio of q-series: A(q) = N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*d^(n^2+n)/( (1 - d^(n+1))*Product_{k = 1..n} (1 - d^k)^2 ) and D(q) = Sum_{n >= 0} (-1)^n*d^(n^2)/( Product_{k = 1..n} (1 - d^k)^2 ).
%F A088354 In the above asymptotic formula of Kotesovec, the constant 1/d = 0.3976807823... is the minimal positive real zero of D(q), and is the dominant singularity of N(q)/D(q). (End)
%t A088354 nmax = 40; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x^k, 1 - x^(k + 1), {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 01 2019 *)
%o A088354 (PARI)
%o A088354 N = 66; x = 'x + O('x^N);
%o A088354 Q(k) = if(k>N, 1,  1 - x^(k+1)*( 1 + 1/Q(k+1) ) );
%o A088354 gf = 1/Q(0);
%o A088354 Vec(gf)
%o A088354 /* _Joerg Arndt_, May 01 2013 */
%Y A088354 Cf. A088352, A224704.
%K A088354 nonn,easy
%O A088354 0,2
%A A088354 _Paul D. Hanna_, Sep 26 2003
%E A088354 Added more terms, _Joerg Arndt_, May 01 2013