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A156281
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Irregular table with the coefficient [x^k] of product_{j=1..n} {x^j-(1-x^j)/(1-x)} in row n, column 0<=k.
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1, -1, 1, 1, 0, -2, 1, -1, -1, 1, 2, 1, -3, 1, 1, 2, 1, -1, -4, -2, 0, 3, 3, -4, 1, -1, -3, -4, -3, 1, 5, 8, 5, -1, -4, -5, -3, 3, 6, -5, 1, 1, 4, 8, 11, 10, 5, -5, -15, -19, -17, -7, 5, 13, 9, 7, -1, -7, -8, 1, 10, -6, 1, -1, -5, -13, -24, -34, -39, -34, -17, 9, 38, 59, 63, 50, 26, -6
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| The row sums are: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,....
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 sub-cyclotomic-Bonacci product where it will work for the q-product.
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LINKS
| L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. Volume 15, Number 4 (1948), 987-1000.
L. Carlitz, J. Riordan, Two element lattice permutation numbers and their q-generalization, Duke Math. J. Volume 31, Number 3 (1964), 371-388
John Shareshian, Michelle L. Wachs, q-Eulerian Polynomials: Excedance Number and Major Index, arXiv:math/0608274 11 Aug 2006, page 3.
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EXAMPLE
| 1;
-1, 1; # -1+x
1, 0, -2, 1; # 1-2*x^2+x^3
-1, -1, 1, 2, 1, -3, 1; # -1-x+x^2+2*x^3+x^4-3*x^5+x^6
1, 2, 1, -1, -4, -2, 0, 3, 3, -4, 1;
-1, -3, -4, -3, 1, 5, 8, 5, -1, -4, -5, -3, 3, 6, -5, 1;
1, 4, 8, 11, 10, 5, -5, -15, -19, -17, -7, 5, 13, 9, 7, -1, -7, -8, 1, 10, -6, 1;
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MATHEMATICA
| Clear[p, n, x]; p[x_, n_] = Product[x^k - (1 - x^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
| Sequence in context: A111335 A163768 A029434 * A002217 A157047 A059342
Adjacent sequences: A156278 A156279 A156280 * A156282 A156283 A156284
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KEYWORD
| tabf,uned,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 07 2009
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