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 A168296 Worpitzky form polynomials for the {1,16,1} A142462 sequence: p(x,n) = Sum[A(n, k)*Binomial[x + k - 1, n - 1], {k, 1, n}] 0
 1, 1, 2, 2, 18, 18, 6, 156, 432, 288, 24, 792, 7416, 13248, 6624, 120, -11280, 64800, 374400, 496800, 198720, 720, -62640, -1254960, 4968000, 20865600, 22057920, 7352640, 5040, 24012000, -11854080, -125677440, 389491200, 1288103040 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Row sums are: {1, 3, 38, 882, 28104, 1123560, 53927280, 3019902480, 193273557120, 13915694298240,...}. 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, 18, 18}, {6, 156, 432, 288}, {24, 792, 7416, 13248, 6624}, {120, -11280, 64800, 374400, 496800, 198720}, {720, -62640, -1254960, 4968000, 20865600, 22057920, 7352640}, {5040, 24012000, -11854080, -125677440, 389491200, 1288103040, 1132306560, 323516160}, {40320, 192378240, 5004581760, -1669248000, -12569437440, 32116331520, 87702289920, 65997296640, 16499324160}, {362880, -119545632000, 57161064960, 868954106880, -218287560960, -1293900894720, 2812649495040, 6545378949120, 4306323605760, 956960801280} MATHEMATICA (*Worpitzky form polynomials for A142462*) Clear[A, m, n, k, a, p] m = 7; 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: A225123 A087338 A055735 * A205454 A100304 A213271 Adjacent sequences:  A168293 A168294 A168295 * A168297 A168298 A168299 KEYWORD uned,sign AUTHOR Roger L. Bagula, Nov 22 2009 STATUS approved

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Last modified April 20 11:52 EDT 2021. Contains 343135 sequences. (Running on oeis4.)