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.

A201159 Irregular triangle read by rows: number of {0,1,2}-shifted Schroeder paths of length n and area k.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 4, 5, 5, 5, 3, 1, 1, 1, 2, 2, 4, 5, 8, 10, 12, 13, 15, 17, 16, 13, 9, 4, 1, 1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 23, 27, 34, 40, 47, 52, 56, 57, 57, 56, 50, 39, 26, 14, 5, 1, 1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 26, 32, 42, 50, 63
Offset: 0

Views

Author

N. J. A. Sloane, Nov 27 2011

Keywords

Examples

			Triangle begins
1
1 1
1 1 2 2 1
1 1 2 2 4 5 5 5 3 1
1 1 2 2 4 5 8 10 12 13 15 17 16 13 9 4 1
...

Links

Crossrefs

Row sums are A064641.
Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201076 {0,2}, A201079 {0,2,4,6...}, A201080 {0,1,3,5...}.

Programs

  • Mathematica
    gf = Expand /@ FixedPoint[1 + x # + q x (1 + q x) # (Normal@# /. {x :> q^2 x}) + O[x]^7 &, 0];
Flatten[Reverse[CoefficientList[#, q]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 03 2024 *)

Extensions

More terms from Andrey Zabolotskiy, Jan 03 2024

A201076 Irregular triangle read by rows: number of {0,2}-shifted Schroeder paths of length n and area k.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 3, 6, 7, 7, 5, 0, 0, 1, 2, 3, 6, 10, 13, 16, 20, 19, 15, 8, 0, 0, 1, 2, 3, 6, 10, 16, 22, 29, 39, 48, 53, 56, 57, 46, 30, 13, 0, 0, 0, 1, 2, 3, 6, 10, 16, 25, 35, 48, 66, 85, 106, 127, 147, 167, 179, 178, 168, 146, 103, 58, 21, 0, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane, Nov 26 2011

Keywords

Examples

			Triangle begins
1
1
1 2 0
1 2 3 3 0
1 2 3 6 7 7 5 0 0
1 2 3 6 10 13 16 20 19 15 8 0 0
...

Links

Crossrefs

Row sums give A052709. Rows converge to A101277.
Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201079 {0,2,4,6...}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

Programs

  • Mathematica
    gf = Expand /@ FixedPoint[1 + (q x + q^2 x^2) # (Normal@# /. {x :> q^2 x}) + O[x]^8 &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Extensions

Row 5 corrected, rows 6-7 added by Andrey Zabolotskiy, Jan 02 2024

A201079 Irregular triangle read by rows: number of {0,2,4,6...}-shifted Schroeder paths of length n and area k.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 4, 6, 7, 7, 5, 0, 0, 1, 2, 4, 7, 11, 14, 18, 20, 19, 15, 8, 0, 0, 1, 2, 4, 8, 12, 19, 26, 35, 43, 52, 57, 61, 57, 46, 30, 13, 0, 0, 0, 1, 2, 4, 8, 13, 21, 32, 45, 61, 81, 101, 125, 146, 167, 183, 194, 191, 178, 146, 103, 58, 21, 0, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane, Nov 26 2011

Keywords

Examples

			Triangle begins
1
1
1 2 0
1 2 3 3 0
1 2 4 6 7 7 5 0 0
1 2 4 7 11 14 18 20 19 15 8 0 0
1 2 4 8 12 19 26 35 43 52 57 61 57 46 30 13 0 0 0
...

Links

Crossrefs

Row sums give A063020. Rows converge to A015128.
Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201076 {0,2}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

Programs

  • Mathematica
    max = 8; s0 = Range[2, max, 2];
gf = Expand /@ FixedPoint[With[{g = Normal@#}, 1 + q x g (g /. {x :> q^2 x}) + Sum[q^(j^2 - j) x^j Product[g /. {x :> q^(2 i - 2) x}, {i, j}], {j, s0}] + O[x]^max] &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Extensions

Name and rows 3 and 5 corrected and row 7 added by Andrey Zabolotskiy, Jan 02 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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