0,6

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.

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.

p(x,n)=Product[x^k - (1 - x^k)/(1 - x), {k, 1, n}].

{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, -28, -36, -25, -9, 2, 13, 17, 8, -5, -14, -4, 15, -7, 1},

{1, 6, 19, 43, 77, 116, 150, 167, 156, 109, 29, -69, -163, -233, -256, -228, -157, -65, 24, 89, 113, 98, 52, 7, -23, -33, -41, -21, 2, 23, 22, 4, -19, -13, 21, -8, 1},

{-1, -7, -26, -69, -146, -262, -412, -579, -735, -842, -860, -759, -529, -185, 226, 638, 979, 1189, 1227, 1087, 807, 452, 97, -189, -366, -419, -351, -217, -65, 49, 110, 112, 93, 53, -11, -52, -54, -29, 15, 36, 22, -20, -27, 28, -9, 1},

{1, 8, 34, 103, 249, 511, 923, 1502, 2237, 3079, 3937, 4683, 5167, 5240, 4791, 3775, 2234, 299, -1819, -3855, -5540, -6650, -7046, -6698, -5695, -4226, -2555, -939, 391, 1289, 1706, 1691, 1340, 812, 280, -140, -369, -423, -351, -224, -75, 59, 160, 145, 79, -15, -71, -75, -16, 41, 49, -13, -47, 36, -10, 1}

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[%]

Sequence in context: A111335 A163768 A029434 this_sequence A002217 A157047 A059342

Adjacent sequences: A156278 A156279 A156280 this_sequence A156282 A156283 A156284

tabl,uned,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 07 2009