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A193560 Augmentation of the Catalan triangle, A009766. See Comments. 3
1, 1, 1, 1, 3, 3, 1, 6, 14, 14, 1, 10, 41, 86, 86, 1, 15, 95, 327, 645, 645, 1, 21, 190, 965, 2991, 5662, 5662, 1, 28, 343, 2410, 10684, 30827, 56632, 56632, 1, 36, 574, 5334, 31969, 128959, 352936, 633545, 633545, 1, 45, 906, 10766, 83860, 449435 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Regarding A193560, if the triangle is written as (w(n,k)), then w(n,n)=A127715(n).

LINKS

Table of n, a(n) for n=0..50.

EXAMPLE

First 5 rows of A193560:

1

1...1

1...3...3

1...6...14...14

1...10..41...86...86

MATHEMATICA

p[n_, k_] := ((n - k + 1)/(n + 1)) (n + k)!/(n!*k!) (* Catalan triangle, A009766 *)

Table[p[n, k], {n, 0, 5}, {k, 0, n}]

m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]

TableForm[m[4]]

w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];

v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};

v[n_] := v[n - 1].m[n]

TableForm[Table[v[n], {n, 0, 6}]] (* A193560 *)

Flatten[Table[v[n], {n, 0, 10}]]

CROSSREFS

Cf. A193091.

Sequence in context: A208524 A094040 A039798 * A278390 A356916 A001498

Adjacent sequences: A193557 A193558 A193559 * A193561 A193562 A193563

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jul 30 2011

STATUS

approved

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Last modified January 28 12:46 EST 2023. Contains 359892 sequences. (Running on oeis4.)