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 A168295 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}] 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. LINKS FORMULA p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}] EXAMPLE {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} MATHEMATICA (*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[%] CROSSREFS Sequence in context: A135996 A141610 A019241 * A309751 A249152 A216708 Adjacent sequences:  A168292 A168293 A168294 * A168296 A168297 A168298 KEYWORD nonn,uned AUTHOR Roger L. Bagula, Nov 22 2009 STATUS approved

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Last modified April 11 05:01 EDT 2021. Contains 342886 sequences. (Running on oeis4.)