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A123221 Bezier transform of Mahonian numbers triangle A008302; p(k, x) = Sum[x^m, {m, 0, k}]*p(k - 1, x). 0
1, 1, 1, 0, 1, 1, 1, 0, 2, 3, 5, 3, 1, 1, 0, 3, 5, 11, 22, 20, 15, 9, 4, 1, 1, 0, 4, 7, 18, 41, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 0, 5, 9, 26, 64, 154, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1, 1, 0, 6, 11, 35, 91, 234, 583, 1415, 1940, 2493 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

This method of recursive polynomials is a better method of obtaining these polynomials than that given originally in A008302. The jump in polynomial power levels in Mahonian numbers is very like that in Poncelet transforms.

LINKS

Table of n, a(n) for n=1..73.

FORMULA

p(k, x) = Sum[x^m, {m, 0, k}]*p(k - 1, x)->t(n,m) Coefficient Bezier transform : t'(n,m)=t(n,m)*x^m*(1-x)^(n-m)

EXAMPLE

{1},

{1},

{1, 0, 1, 1},

{1, 0, 2, 3, 5, 3, 1},

{1, 0, 3, 5, 11, 22, 20, 15, 9, 4, 1},

{1, 0, 4, 7, 18, 41, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1},

{1,0, 5, 9, 26, 64, 154, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98,

49, 20, 6, 1},

MATHEMATICA

(* Mahonian Polynomials*) p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = Sum[x^m, {m, 0, k}]*p[k - 1, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; (* Bezier transform*) v = Table[CoefficientList[Sum[w[[n + 1]][[m + 1]]*x^m*(1 - x)^(n - m), {m, 0, Length[w[[n + 1]]] - 1}], x], {n, 0, 10}]; Flatten[v]

CROSSREFS

Cf. A008302.

Sequence in context: A069110 A238684 A202694 * A197032 A172984 A072751

Adjacent sequences:  A123218 A123219 A123220 * A123222 A123223 A123224

KEYWORD

nonn,uned,tabf

AUTHOR

Roger L. Bagula, Oct 05 2006

STATUS

approved

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Last modified December 20 20:08 EST 2014. Contains 252289 sequences.