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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

Table of n, a(n) for n=0..84.

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 A242357 A120423

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 August 23 22:45 EDT 2019. Contains 326254 sequences. (Running on oeis4.)