cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A114656 Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 24, 12, 1, 16, 80, 80, 20, 1, 32, 240, 400, 200, 30, 1, 64, 672, 1680, 1400, 420, 42, 1, 128, 1792, 6272, 7840, 3920, 784, 56, 1, 256, 4608, 21504, 37632, 28224, 9408, 1344, 72, 1, 512, 11520, 69120, 161280, 169344, 84672, 20160, 2160, 90
Offset: 1

Views

Author

Emeric Deutsch, Dec 23 2005

Keywords

Comments

Row sums are the little Schroeder numbers (A001003). Sum_{k=1..n} k*T(n,k) = A047781(n). T(n,k) = (1/2)A114655(n,k).
Triangle T(n,k), 1 <= k <= n, given by [0,2,0,2,0,2,0,2,0,2,0,2,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 02 2009

Examples

			T(3,2)=6 because we have (UD)Ub(UD)D, (UD)Ur(UD)D, Ub(UD)D(UD), Ur(UD)D(UD), Ub(UD)(UD)D and Ur(UD)(UD)D, where U=(1,1), D=(1,-1) and b (r) indicates a blue (red) double rise (the peaks are shown between parentheses).
Triangle begins:
   1;
   2,  1;
   4,  6,  1;
   8, 24, 12,  1;
  16, 80, 80, 20,  1;
  ....
Triangle T(n,k), 0 <= k <= n, given by [0,2,0,2,0,2,0,2,...] DELTA [1,0,1,0,1,0,1,0,1,0,...] begins: 1; 0,1; 0,2,1; 0,4,6,1; 0,8,24,12,1; 0,16,80,80,20,1; ... - _Philippe Deléham_, Jan 02 2009

Links

Crossrefs

Cf. A001003, A047781, A114655.

Programs

  • Maple
    T:=(n,k)->2^(n-k)*binomial(n,k)*binomial(n,k-1)/n: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
  • Mathematica
    Table[2^(n - k) Binomial[n, k] Binomial[n, k - 1]/n, {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Apr 23 2019 *)

Formula

T(n, k) = 2^(n-k)*binomial(n, k)*binomial(n, k-1)/n.
G.f.: G = G(t, z) satisfies G = z(2G+t)(G+1).
T(n,k) = A001263(n,k)*2^(n-k). - Philippe Deléham, Apr 11 2007
G.f.: 1/(1-xy/(1-2x/(1-xy/(1-2x/(1-xy/(1-2x/(1-..... (continued fraction). - Paul Barry, Feb 06 2009

A114690 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k weak ascents (1 <= k <= ceiling(n/2)).

Original entry on oeis.org

1, 2, 3, 1, 5, 4, 8, 12, 1, 13, 31, 7, 21, 73, 32, 1, 34, 162, 116, 11, 55, 344, 365, 70, 1, 89, 707, 1041, 335, 16, 144, 1416, 2762, 1340, 135, 1, 233, 2778, 6932, 4726, 820, 22, 377, 5358, 16646, 15176, 4039, 238, 1, 610, 10188, 38560, 45305, 17157, 1785, 29, 987
Offset: 1

Views

Author

Emeric Deutsch, Dec 24 2005

Keywords

Comments

A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis. A weak ascent in a Motzkin path is a maximal sequence of consecutive U and H steps.
Row n has ceiling(n/2) terms.
Row sums are the Motzkin numbers (A001006).
Column 1 yields the Fibonacci numbers (A000045).
Sum_{k=1..ceiling(n/2)} k*T(n,k) = A005773(n).

Examples

			T(4,2)=4 because we have (HU)D(H),(U)D(HH),(U)D(U)D and (UH)D(H) (the weak ascents are shown between parentheses).
Triangle starts:
   1;
   2;
   3,  1;
   5,  4;
   8, 12,  1;
  13, 31,  7;
  ...

Links

Crossrefs

Cf. A001006, A005773, A000045, A114655.

Programs

  • Maple
    G:=(1-t*z^2-z-z^2-sqrt(1-2*t*z^2-2*z-z^2+t^2*z^4-2*t*z^3-2*z^4*t+2*z^3+z^4))/2/z^2: Gser:=simplify(series(G,z=0,18)): for n from 1 to 15 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..ceil(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, t,
      b(x-1, y+1, z)+expand(b(x-1, y-1, 1)*t)+b(x-1, y, z)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..degree(p)))(b(n, 0, 1)):
seq(T(n), n=1..14);  # Alois P. Heinz, Nov 16 2019
  • Mathematica
    b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, t,
     b[x-1, y+1, z] + Expand[b[x-1, y-1, 1]*t] + b[x-1, y, z]]];
T[n_] := CoefficientList[b[n, 0, 1]/z, z];
Array[T, 14] // Flatten (* Jean-François Alcover, Feb 14 2021, after Alois P. Heinz *)

Formula

G.f. G = G(t, z) satisfies G = z*(t+G)*(1+z+z*G).

A114691 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k weak ascents (1<=k<=n-1 for n>=2; k=1 for n=1). A Schroeder path of length 2n 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. A hill is a peak at height 1. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.

Original entry on oeis.org

1, 3, 7, 4, 15, 26, 4, 31, 108, 54, 4, 63, 366, 380, 90, 4, 127, 1104, 1950, 960, 134, 4, 255, 3090, 8284, 6966, 2008, 186, 4, 511, 8212, 31030, 39780, 19550, 3716, 246, 4, 1023, 21014, 106252, 192802, 144472, 46670, 6308, 314, 4, 2047, 52248, 340190
Offset: 1

Views

Author

Emeric Deutsch, Dec 24 2005

Keywords

Comments

Row n contains n-1 terms (n>=2). Row sums are the little Schroeder numbers (A001003).

Examples

			T(3,2)=4 because we have (UH)D(H),(UU)DD(H),(UU)D(H)D and (UU)D(U)DD, where U=(1,1), D=(1,-1) and H=(2,0) (the weak ascents are shown between parentheses).
Triangle starts:
1;
3;
7,4;
15,26,4;
31,108,54,4;

Crossrefs

Cf. A114655.

Programs

  • Maple
    H:=(1-z*t-2*z-sqrt(1-2*z*t-4*z+z^2*t^2-4*z^2*t+4*z^2))/2/z: G:=z*(t+H)/(1-z-z*H): Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: 1; for n from 2 to 11 do seq(coeff(P[n],t^j),j=1..n-1) od; # yields sequence in triangular form

Formula

G.f.=G=z(t+H)/(1-z-zH), where H is given by H =z(2+H)(t+H).
Showing 1-3 of 3 results.

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