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A203717 A Catalan triangle by rows. 1
1, 1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 20, 15, 5, 1, 1, 50, 53, 21, 6, 1, 1, 126, 182, 84, 28, 7, 1, 1, 322, 616, 326, 120, 36, 8, 1, 1, 834, 2070, 1242, 495, 165, 45, 9, 1, 1, 2187, 6930, 4680, 1997, 715, 220, 55, 10, 1, 1, 5797, 23166, 17512, 7942, 3003, 1001, 286, 66, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums = the Catalan sequence starting with offset 1: (1, 2, 5, 14, 42,...).

T(n,k) is the number of Dyck n-paths whose maximum ascent length is k. - David Scambler, Aug 22 2012

T(n,k) is the number of ordered rooted trees with n non-root nodes and maximal outdegree k. T(4,3) = 4:

.    o      o      o      o

.    |     /|\    /|\    /|\

.    o    o o o  o o o  o o o

.   /|\   |        |        |

.  o o o  o        o        o   - Alois P. Heinz, Jun 29 2014

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

Finite differences of antidiagonals of an array in which n-th array row is generated from powers of M, extracting successive upper left terms. M for n-th row of the array is an infinite square production matrix composed of (n+1) diagonals of 1's and the rest zeros. Given the upper left term of the array is (1,1), the diagonals begin at (1,2), (1,1), (2,1), (3,1), (4,1),...

EXAMPLE

First few rows of the array begin:

1,...1,...1,...1,...1,...;

1,...2,...4,...9,..21,...; = A001006

1,...2,...5,..13,..36,...; = A036765

1,...2,...5,..14,..41,...; = A036766

1,...2,...5,..14,..42,...; = A036767

... Taking finite differences of array terms starting from the top by columns, we obtain row terms of the triangle. First few rows of the triangle are:

1;

1, 1;

1, 3, 1;

1, 8, 4, 1;

1, 20, 15, 5, 1;

1, 50, 53, 21, 6, 1;

1, 126, 182, 84, 28, 7, 1;

1, 322, 616, 326, 120, 36, 8, 1;

1, 834, 2070, 1242, 495, 165, 45, 9, 1;

1, 2187, 6930, 4680, 1997, 715, 220, 55, 10, 1;

...

Example: Row 4 of the triangle = (1, 8, 4, 1) = the finite differences of (1, 9, 13, 14), column 4 of the array. Term (3,4) = 13 of the array is the upper left term of M^4, where M is an infinite square production matrix with four diagonals of 1's starting at (1,2), (1,1), (2,1), and (3,1); with the rest zeros.

MAPLE

b:= proc(n, t, k) option remember; `if`(n=0, 1, `if`(t>0,

      add(b(j-1, k$2)*b(n-j, t-1, k), j=1..n), b(n-1, k$2)))

    end:

T:= (n, k)-> b(n, k-1$2) -`if`(k=1, 0, b(n, k-2$2)):

seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Jun 29 2014

MATHEMATICA

b[n_, t_, k_] := b[n, t, k] = If[n == 0, 1, If[t > 0, Sum[b[j-1, k, k]*b[n - j, t-1, k], {j, 1, n}], b[n-1, k, k]]]; T[n_, k_] := b[n, k-1, k-1] - If[k == 1, 0, b[n, k-2, k-2]]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, May 27 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A000108, A001006, A036765, A036766, A036767.

Sequence in context: A121461 A273719 A274488 * A143953 A114276 A152879

Adjacent sequences:  A203714 A203715 A203716 * A203718 A203719 A203720

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jan 04 2012

STATUS

approved

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Last modified April 27 00:46 EDT 2017. Contains 285506 sequences.