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A168295
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Worpitzky form polynomials for the {1,8,1} A142458 sequence: p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}]
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0
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1, 1, 2, 2, 10, 10, 6, 52, 120, 80, 24, 280, 1160, 1760, 880, 120, 1520, 10000, 27200, 30800, 12320, 720, 11280, 78160, 343200, 695200, 628320, 209440, 5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800, 40320, 1438080
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OFFSET
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1,3
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COMMENTS
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Row sums are:
{1, 3, 22, 258, 4104, 81960, 1966320, 55051920, 1761621120, 63417997440,...}.
Dividing row A167786 by 3^n gets a very similar sequence.
In Comtet there is this function:
x^n=Sum[Eulerian[n,k*Binomial[x+k-1,n],{k,1,n]]
In OEIS I was looking for an Umbral Calculus expansion for the MacMahon and
found this "Worpitzky form":
Sum [MacMahon[n,k]*Binomial[x+k-1,n-1],{k,1,n}]=(2*x+1)^(n+1)
The use the infinite sums k, 2*k+1 type polynomials
and are pretty much alike except for a sliding offset in n.
Conjecture: "Worpitzky forms"
Some general polynomial form:general Pascal recursion Pascal[n,k,m]
p[x,n,m]=Sum [Pascal[n,k,m]*Binomial[x+k-1,n-1],{k,1,n}]
where p[x,n,m] are the inverse z transform polynomials.
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LINKS
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Table of n, a(n) for n=1..38.
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FORMULA
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p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}]
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EXAMPLE
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{1},
{1, 2},
{2, 10, 10},
{6, 52, 120, 80},
{24, 280, 1160, 1760, 880},
{120, 1520, 10000, 27200, 30800, 12320},
{720, 11280, 78160, 343200, 695200, 628320, 209440},
{5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800},
{40320, 1438080, 15532160, 48294400, 170755200, 445688320, 598160640, 385369600, 96342400},
{362880, -51206400, 178617600, 1217036800, 2840745600, 8032738560, 17417030400, 20005708800, 11272060800, 2504902400}
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MATHEMATICA
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(*Worpitzky form polynomials for A142458*)
Clear[A, m, n, k, a, p]
m = 3;
A[n_, 1] := 1 A[n_, n_] := 1
A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 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[%]
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CROSSREFS
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A167786, A142458
Sequence in context: A135996 A141610 A019241 * A309751 A249152 A216708
Adjacent sequences: A168292 A168293 A168294 * A168296 A168297 A168298
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula, Nov 22 2009
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STATUS
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approved
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