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A156282
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A triangle sequence of cyclotomic product polynomials: p(x,n)=Product[Cyclotomic[k + 1, x], {k, 1, n}].
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0
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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
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OFFSET
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0,5
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COMMENTS
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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.
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REFERENCES
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L. Carlitz,q-Bernoulli numbers and polynomials,Duke Math. J. Volume 15, Number 4 (1948), 987-1000.http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077475200
L. Carlitz and J. Riordan,Two element lattice permutation numbers and their q-generalization, Duke Math. J. Volume 31, Number 3 (1964), 371-388, http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077375351
John Shareshian, Michelle L. Wachs, q-Eulerian Polynomials : Excedance Number ans Major Index, arXiv: math/ 0608274v1,11 Aug 2006,page 3.
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LINKS
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Table of n, a(n) for n=0..84.
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FORMULA
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p(x,n)=Product[Cyclotomic[k + 1, x], {k, 1, n}].
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EXAMPLE
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{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},
{1, 2, 5, 9, 16, 25, 38, 53, 72, 92, 114, 135, 155, 171, 183, 189, 189, 183, 171, 155, 135, 114, 92, 72, 53, 38, 25, 16, 9, 5, 2, 1},
{1, 3, 8, 17, 33, 58, 96, 149, 221, 313, 427, 561, 714, 880, 1054, 1227, 1391, 1536, 1654, 1737, 1780, 1780, 1737, 1654, 1536, 1391, 1227, 1054, 880, 714, 561, 427, 313, 221, 149, 96, 58, 33, 17, 8, 3, 1}
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MATHEMATICA
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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[%]
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CROSSREFS
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Sequence in context: A136605 A165621 A004739 * A203776 A120423 A113137
Adjacent sequences: A156279 A156280 A156281 * A156283 A156284 A156285
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KEYWORD
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nonn,tabl,uned
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AUTHOR
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Roger L. Bagula, Feb 07 2009
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STATUS
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approved
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