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A123221
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Bezier transform of Mahonian numbers triangle A008302; p(k, x) = Sum[x^m, {m, 0, k}]*p(k - 1, x).
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0
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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
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OFFSET
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1,9
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..73.
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FORMULA
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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)
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EXAMPLE
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{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},
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MATHEMATICA
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(* 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]
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CROSSREFS
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Cf. A008302.
Sequence in context: A113195 A069110 A202694 * A197032 A172984 A072751
Adjacent sequences: A123218 A123219 A123220 * A123222 A123223 A123224
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KEYWORD
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nonn,uned,tabf
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AUTHOR
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Roger Bagula, Oct 05 2006
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STATUS
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approved
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