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/(...))))).

1, 2, 4, 10, 24, 60, 150, 376, 944, 2372, 5962, 14988, 37684, 94752, 238252, 599090, 1506440, 3788036, 9525280, 23952020, 60229184, 151450970, 380835368, 957640640, 2408063340, 6055266600, 15226449480, 38288118984, 96278523274, 242100012876, 608779761460, 1530825191912

Offset: 0

Author

Paul D. Hanna, Sep 26 2003

Keywords

• nonn
• easy

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.

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)

Links

• P. Bala, Illustration for a(3) = 10.
• P. Bala, Triangle Stacks of large Schröder type

Crossrefs

Cf. A088352, A224704.

Programs

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 *)

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 */

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)

Extensions

Added more terms, Joerg Arndt, May 01 2013