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 A156282 A triangle sequence of cyclotomic product polynomials: p(x,n)=Product[Cyclotomic[k + 1, x], {k, 1, n}]. 0
 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 2, 1, 1, 3, 6, 9, 11, 11, 9, 6, 3, 1, 1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1, 1, 3, 7, 13, 21, 30, 39, 46, 50, 50, 46, 39, 30, 21, 13, 7, 3, 1, 1, 3, 7, 13, 22, 33, 46, 59, 71, 80, 85, 85, 80, 71, 59, 46, 33, 22, 13, 7, 3, 1, 1, 3, 7, 14, 25, 40, 60, 84, 111, 139 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The row sums are:{1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720,...}. This idea for these products come from the q-Eulerian exponential generalization: Here are the definitions of the q-exponentials in John Shareshian and Michelle Wachs paper: Clear[Q, e, p, n, x]; p[x_, n_] := Product[(1 - x^k)/(1 - x), {k, 1, n}]; e[x_, q_] = Sum[x^n/p[q, n], {n, 0, Infinity}]; f[x_, t_, q_] = (1 - t)/(e[x*(t - 1), q] - t); Where the expansion is: (called the Stanley q-analog of the Eulerian type); (1 - t)/(e[x*(t - 1), q] - t)= Sum[A[n,q,t]*x^n/p[q, n], {n, 0, Infinity}] ; The idea here is to substitute the new Cyclotomic product where it will work for the q-product. LINKS L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. Volume 15, Number 4 (1948), 987-1000. L. Carlitz and J. Riordan, Two element lattice permutation numbers and their q-generalization, Duke Math. J. Volume 31, Number 3 (1964), 371-388 Y.-H. He, C. Matti, C. Sun, The Scattering Variety, arXiv preprint arXiv:1403.6833 [cs.SE], 2014. See Table 2, central column. - N. J. A. Sloane, Jun 28 2014 John Shareshian, Michelle L. Wachs, q-Eulerian Polynomials: Excedance Number ans Major Index, arXiv:math/0608274 [math.CO], 2006, page 3. FORMULA p(x,n)=Product[Cyclotomic[k + 1, x], {k, 1, n}]. EXAMPLE 1; 1, 1; 1, 2, 2, 1; 1, 2, 3, 3, 2, 1; 1, 3, 6, 9, 11, 11, 9, 6, 3, 1; 1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1; 1, 3, 7, 13, 21, 30, 39, 46, 50, 50, 46, 39, 30, 21, 13, 7, 3, 1; 1, 3, 7, 13, 22, 33, 46, 59, 71, 80, 85, 85, 80, 71, 59, 46, 33, 22, 13, 7, 3, 1; 1, 3, 7, 14, 25, 40, 60, 84, 111, 139, 166, 189, 206, 215, 215, 206, 189, 166, 139, 111, 84, 60, 40, 25, 14, 7, 3, 1 MATHEMATICA Clear[p, n, x]; p[x_, n_] = Product[Cyclotomic[k + 1, x], {k, 1, n}]; Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%] PROG (PARI) row(n) = Vec(prod(k=1, n, polcyclo(k+1))); \\ Michel Marcus, Dec 12 2017 CROSSREFS Sequence in context: A165621 A284321 A004739 * A203776 A343559 A242357 Adjacent sequences:  A156279 A156280 A156281 * A156283 A156284 A156285 KEYWORD nonn,tabf,uned AUTHOR Roger L. Bagula, Feb 07 2009 STATUS approved

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Last modified May 12 10:54 EDT 2021. Contains 343821 sequences. (Running on oeis4.)