A326676 -id:A326676 - cp's OEIS frontend

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

Paul D. Hanna, Sep 26 2003

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)

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

Cf. A088352, A224704.

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 *)

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 */

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)

Added more terms, Joerg Arndt, May 01 2013

A129179 Triangle read by rows: T(n, k) is the number of Schroeder paths of semilength n such that the area between the x-axis and the path is k (n >= 0; 0 <= k <= n^2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 3, 4, 3, 2, 1, 1, 1, 1, 4, 6, 7, 10, 11, 10, 9, 8, 7, 5, 4, 3, 2, 1, 1, 1, 1, 5, 10, 14, 21, 28, 31, 33, 34, 34, 31, 27, 25, 22, 17, 14, 13, 10, 7, 5, 4, 3, 2, 1, 1, 1, 1, 6, 15, 25, 40, 60, 77, 92, 106, 117, 122, 121, 120, 116, 107, 98, 91, 82, 71, 62, 54, 45

Offset: 0

Emeric Deutsch, Apr 08 2007

A Schroeder path of semilength n is a lattice path from (0,0) to (2n,0) consisting of U = (1,1), D = (1,-1) and H = (2,0) steps and never going below the x-axis.
Row n has 1+n^2 terms.
Row sums are the large Schroeder numbers (A006318).

			T(3,5) = 3 because we have UDUUDD, UUDDUD and UHHD.
Triangle starts:
  1;
  1,1;
  1,2,1,1,1;
  1,3,3,3,4,3,2,1,1,1;
  1,4,6,7,10,11,10,9,8,7,5,4,3,2,1,1,1;

Alois P. Heinz, Rows n = 0..32, flattened

Cf. A006318 (row sums), A129180, A326676.

G:=1/(1-z-t*z*g[1]): for i from 1 to 11 do g[i]:=1/(1-t^(2*i)*z-t^(2*i+1)*z*g[i+1]) od: g[12]:=0: Gser:=simplify(series(G,z=0,13)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 6 do seq(coeff(P[n],t,j),j=0..n^2) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, expand(b(x-1, y-1)*z^(y-1/2)
      +b(x-2, y)*z^(2*y) +b(x-1, y+1)*z^(y+1/2))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0)):
seq(T(n), n=0..7);  # Alois P. Heinz, May 27 2015

b[x_, y_] := b[x, y] = If[y>x || y<0, 0, If[x==0, 1, Expand[b[x-1, y-1]*z^(y-1/2)  + b[x-2, y]*z^(2*y) + b[x-1, y+1]*z^(y+1/2)]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0]]; Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)

G.f.: G(t,z) satisfies G(t,z) = 1 + z*G(t,z) + t*z*G(t,t^2*z)*G(t,z).
Sum_{k>=0} k*T(n,k) = A129180(n).
From Peter Bala, Aug 02 2019: (Start)
O.g.f. as a continued fraction: (t marks the area and z marks the semilength of the path)
G(t,z) = 1/(1 - z - t*z/(1 - t^2*z - t^3*z/(1 - t^4*z - t^5*z/(1 - t^6*z - (...) )))) = 1 + z*(1 + t) + z^2*(1 + 2*t + t^2 + t^3 + t^4) + ....
G(t,z) = 1/(1 - (1 + t)*z/(1 - t^3*z/(1 - (t^2 + t^5)*z/(1 - t^7*z/(1 - (t^4 + t^9)*z/(1 - t^11*z/( (...) ))))))).
O.g.f. as a ratio of q-series: N(t,z)/D(t,z), where N(t,z) = Sum_{n >= 0} (-1)^n*t^(2*n^2+n)*z^n/( (Product_{k = 1..n} 1 - t^(2*k)) * (Product_{k = 1..n+1} 1 - t^(2*k-2)*z) ) and D(t,z) = Sum_{n >= 0} (-1)^n*t^(2*n^2-n)*z^n/( (Product_{k = 1..n} 1 - t^(2*k)) * (Product_{k = 1..n} 1 - t^(2*k-2)*z) ). (End)
Conjecture: T(n, k) = [z^k] R(n, 0) for n >= 0, k >= 0 where R(n, q) = Sum_{j=0..q + (q mod 2) + 1} z^j*R(n-1, j) for n > 0, q >= 0 with R(0, q) = 1 for q >= 0. - Mikhail Kurkov, Aug 03 2023
Conjecture above can be simplified as follows: T(n, k) = [z^k] P(n+1, n+1) for n >= 0, k >= 0 where P(n, k) = P(n-1, k) + z^(n-k)*P(n + ((n-k) mod 2), k-1) for 0 < k <= n with P(n, k) = 0 for k > n, P(n, 0) = 1 for n >= 0. - Mikhail Kurkov, Oct 02 2024

cp's OEIS Frontend

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/(...))))).

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