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A168391 Worpitzky form polynomials for the Narayana triangle A001263(n,k):p(x,n) = Sum[A001263(n,k)*Binomial[x + k - 1, n - 1], {k, 1, n}] 0
1, 1, 2, 2, 5, 5, 6, 19, 21, 14, 24, 84, 126, 84, 42, 120, 468, 750, 720, 330, 132, 720, 2988, 5496, 5445, 3795, 1287, 429, 5040, 22356, 43120, 50435, 35035, 19019, 5005, 1430, 40320, 186912, 391688, 472472, 398398, 208208, 92092, 19448, 4862, 362880 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row sums are:

{1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400,...}.

Z transform of the polynomial is the narayan triangle back:

Table[CoefficientList[FullSimplify[ExpandAll[((z - 1)^n/z)*ZTransform[ p[x, n], x, z]]], z], {n, 1, 10}].

Every other polynomial has a factor of (1+2*x), just like the higher Sierpinski -Pascal Worpitzky form polynomials.

LINKS

Table of n, a(n) for n=1..46.

FORMULA

p(x,n) = Sum[A001263(n,k)*Binomial[x + k - 1, n - 1], {k, 1, n}]

EXAMPLE

{1},

{1, 2},

{2, 5, 5},

{6, 19, 21, 14},

{24, 84, 126, 84, 42},

{120, 468, 750, 720, 330, 132},

{720, 2988, 5496, 5445, 3795, 1287, 429},

{5040, 22356, 43120, 50435, 35035, 19019, 5005, 1430},

{40320, 186912, 391688, 472472, 398398, 208208, 92092, 19448, 4862},

{362880, 1762848, 3831048, 5103592, 4358718, 2842476, 1169532, 434928, 75582, 16796}

MATHEMATICA

Clear[A, m, n, k, a, p]

A[n_, k_] = Binomial[n - 1, k - 1]*Binomial[n, k - 1]/k

a = Table[A[n, k], {n, 10}, {k, n}];

p[x_, n_] = Sum[a[[n, k]]*Binomial[x + k - 1, n - 1], {k, 1, n}];

Table[CoefficientList[Expand[(n - 1)!*p[x, n]], x], {n, 1, 10}];

Flatten[%]

CROSSREFS

Cf. A001263

Sequence in context: A069896 A053246 A219651 * A157123 A265764 A213032

Adjacent sequences:  A168388 A168389 A168390 * A168392 A168393 A168394

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Nov 24 2009

STATUS

approved

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Last modified June 20 01:36 EDT 2019. Contains 324223 sequences. (Running on oeis4.)