A061667 -id:A061667 - cp's OEIS frontend

A131249 A007318 * A052509.

1, 2, 1, 4, 4, 1, 8, 12, 5, 1, 16, 32, 18, 6, 1, 32, 80, 57, 24, 7, 1, 64, 192, 168, 82, 31, 8, 1, 128, 448, 471, 260, 113, 39, 9, 1, 256, 1024, 1270, 790, 374, 152, 48, 10, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A061667: (1, 3, 9, 26, 73, 201, ...).

			First few rows of the triangle:
   1;
   2,  1;
   4,  4,  1;
   8, 12,  5,  1;
  16, 32, 18,  6,  1;
  32, 80, 57, 24,  7,  1;
  ...

Cf. A061667, A007318, A052509.

Binomial transform of A052509.

1, 2, 1, 4, 4, 1, 8, 11, 6, 1, 16, 26, 22, 8, 1, 32, 57, 64, 37, 10, 1, 64, 120, 163, 130, 56, 12, 1, 128, 247, 382, 386, 232, 79, 14, 1, 256, 502, 848, 1024, 794, 378, 106, 16, 1, 512, 1013, 1816, 2510, 2380, 1471, 576, 137, 18, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A061667: (1, 3, 9, 26, 73, 201, ...).
Companion triangle = A131249 = A007318 * A052509, where A052509 is the reversal of A004070.
Reversal of A097750. - Philippe Deléham, Jan 11 2014
Riordan array (1/(1-2x), x/(1-x)^2). - Philippe Deléham, Jan 11 2014
Diagonal sums are A045623. - Philippe Deléham, Jan 11 2014

			First few rows of the triangle:
   1;
   2,  1;
   4,  4,  1;
   8, 11,  6,  1;
  16, 26, 22,  8,  1;
  32, 57, 64, 37, 10,  1;
  ...

Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2 (see Table 2).

Cf. A004070, A061667, A131249, A007318.

Binomial transform of A004070.

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014

More terms from Philippe Deléham, Jan 11 2014

A322378 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck prefixes (i.e., left factors of nondecreasing Dyck paths) of length n and final height k (0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 13, 0, 14, 0, 6, 0, 1, 13, 0, 26, 0, 20, 0, 7, 0, 1, 0, 34, 0, 45, 0, 27, 0, 8, 0, 1, 34, 0, 73, 0, 71, 0, 35, 0, 9, 0, 1, 0, 89, 0, 137, 0, 105, 0, 44, 0, 10, 0, 1, 89, 0, 201, 0, 234, 0, 148, 0, 54, 0, 11, 0, 1, 0, 233, 0, 402, 0, 373, 0, 201, 0, 65, 0, 12, 0, 1, 233, 0, 546, 0, 733, 0, 564, 0, 265, 0, 77, 0, 13, 0, 1, 0, 610, 0, 1149, 0, 1245, 0, 818, 0, 341, 0, 90, 0, 14, 0, 1

Offset: 0

José Luis Ramírez Ramírez, Dec 05 2018

			Triangle begins:
   1;
   0,   1;
   1,   0,   1;
   0,   2,   0,   1;
   2,   0,   3,   0,   1;
   0,   5,   0,   4,   0,   1;
   5,   0,   9,   0,   5,   0,   1;
   0,  13,   0,  14,   0,   6,   0,   1;
  13,   0,  26,   0,  20,   0,   7,   0,   1;
   0,  34,   0,  45,   0,  27,   0,   8,   0,   1;
  34,   0,  73,   0,  71,   0,  35,   0,   9,   0,   1;
   0,  89,   0, 137,   0, 105,   0,  44,   0,  10,   0,   1;
  89,   0, 201,   0, 234,   0, 148,   0,  54,   0,  11,   0,   1;
   0, 233,   0, 402,   0, 373,   0, 201,   0,  65,   0,  12,   0,   1;
  ...

R. Flórez and J. L. Ramírez, Some enumerations on non-decreasing Motzkin paths, Australasian Journal of Combinatorics, 72(1) (2018), 138-154.

Columns k=0, 1 give A001519. Column k=2 gives A061667.

Cf. A322329, A322325.

Riordan array: ((1 - 2*x^2)/(1 - 3*x^2 + x^4), (x*(1-x^2))/(1 - 2*x^2)).

cp's OEIS Frontend

A131249 A007318 * A052509.

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A131250 A007318 * A004070.

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A322378 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck prefixes (i.e., left factors of nondecreasing Dyck paths) of length n and final height k (0 <= k <= n).

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Keywords

Examples

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