A369326 -id:A369326 - cp's OEIS frontend

A369328 Antidiagonal sums of A369326.

1, 2, 4, 9, 23, 63, 176, 491, 1362, 3762, 10369, 28559, 78653, 216638, 596761, 1643966, 4528932, 12476737, 34372059, 94691059, 260862508, 718644363, 1979777046, 5454042866, 15025219777, 41392641187, 114031662049, 314143279634, 865426307713, 2384143611738, 6568023995156

Offset: 0

Stefano Spezia, Jan 20 2024

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 22.
Index entries for linear recurrences with constant coefficients, signature (5,-8,5).

Cf. A369326.

First column of A372883.

Mathematica
```
LinearRecurrence[{5,-8,5},{1,1,2,4},31]
```

G.f.: (1 - 4*x + 5*x^2 - 3*x^3)/(1 - 5*x + 8*x^2 - 5*x^3).

a(n) = 5*a(n-1) - 8*a(n-2) + 5*a(n-3) for n > 3.

A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 16, 25, 16, 5, 1, 0, 1, 32, 65, 56, 25, 6, 1, 0, 1, 64, 161, 176, 105, 36, 7, 1, 0, 1, 128, 385, 512, 385, 176, 49, 8, 1, 0, 1, 256, 897, 1408, 1281, 736, 273, 64, 9, 1, 0, 1, 512, 2049, 3712, 3969, 2752, 1281, 400, 81, 10, 1

Offset: 0

Stefano Spezia, Jan 20 2024

			The array begins:
  0, 1,  1,   1,   1,    1, ...
  0, 1,  2,   3,   4,    5, ...
  0, 1,  4,   9,  16,   25, ...
  0, 1,  8,  25,  56,  105, ...
  0, 1, 16,  65, 176,  385, ...
  0, 1, 32, 161, 512, 1281, ...
  ...

Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See Theorem 3.18 at page 10.

Cf. A000004 (k=0), A000012 (k=1), A000079 (k=2), A002064 (k=3), A340257 (k=4).
Cf. A000290 (n=2), A001477 (n=1), A057427 (n=0), A131423 (n=3), A164039.
Cf. A000035, A369325 (main diagonal), A369326.

A[n_,k_]:=(1-(-1)^k)/2+2^n Sum[Binomial[n+k-3-2i,n-1],{i,0,Floor[(k-2)/2]}]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

A(n,k) = A000035(k) + 2^n*Sum_{i=0..floor((k-2)/2)} binomial(n + k - 3 - 2*i, n - 1).

Sum_{k=0..n} A(n-k,k) = A164039(n-1).

A369327 a(n) is the number of words of length n over the alphabet [n] and sortable by a (2,1)-pop stack of depth 2.

Original entry on oeis.org

1, 1, 4, 24, 149, 923, 5784, 36708, 235156, 1516864, 9838018, 64093998, 419137378, 2749637566, 18087479624, 119263349200, 788015054253, 5216212564723, 34584497597602, 229635524379026, 1526740206967913, 10162620233598959, 67719554546286284, 451699650863065424

Offset: 0

Stefano Spezia, Jan 20 2024

nonn

Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See Theorem 3.25 at page 13.

Main diagonal of A369326.

Cf. A369325.

A369326[n_,k_]:=SeriesCoefficient[((1-x)(1-2x)-((1-x)(1-2x)+x^2)y)/((1-x)(1-2x)-(1-x)(2-3x)y+(1-2x)y^2),{x,0,n},{y,0,k}]; a[n_]:=A369326[n,n]; Array[a,24,0]

a(n) = A369326(n, n).

cp's OEIS Frontend

A369328 Antidiagonal sums of A369326.

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A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.

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Formula

A369327 a(n) is the number of words of length n over the alphabet [n] and sortable by a (2,1)-pop stack of depth 2.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula