OFFSET
0,2
COMMENTS
From Peter Bala, Jul 29 2019: (Start)
a(n) is the number of triangle stacks of large Schröder type containing n down-triangles. See Links for a definition and an illustration.
LINKS
FORMULA
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
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
a(n) ~ c * d^n, where d = 2.514579643878729188510437194343099820141030855900783271935495710723840992... and c = 0.589519721244409964128200577034763735132770782513329859477444288778116... - Vaclav Kotesovec, Jul 01 2019
From Peter Bala, Jul 29 2019: (Start)
O.g.f. as a continued fraction:
1/(1 - 2*d/(1 - d^2/(1 - (d^2 + d^3)/(1 - d^4/(1 - (d^3 + d^5)/(1 - d^6/( (...) ))))))).
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 ).
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)
MATHEMATICA
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 *)
PROG
(PARI)
N = 66; x = 'x + O('x^N);
Q(k) = if(k>N, 1, 1 - x^(k+1)*( 1 + 1/Q(k+1) ) );
gf = 1/Q(0);
Vec(gf)
/* Joerg Arndt, May 01 2013 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Sep 26 2003
EXTENSIONS
Added more terms, Joerg Arndt, May 01 2013
STATUS
approved