A156282 Irregular triangle read by rows: row n is the expansion of the product of cyclotomic polynomials from 2nd to n-th polynomial.

1, 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

Offset

0,6

Links

Alois P. Heinz, Rows n = 0..60, flattened

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, and 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 and Michelle L. Wachs, q-Eulerian Polynomials: Excedance Number and Major Index, arXiv:math/0608274 [math.CO], 2006, page 3.

Example

1;
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

Maple

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
         mul(numtheory[cyclotomic](i, x), i=2..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 15 2022

Mathematica

row[n_] := CoefficientList[Product[Cyclotomic[k+1, x], {k, 1, n}], x];
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jul 15 2022 *)

Prog

(PARI) row(n) = Vec(prod(k=1, n, polcyclo(k+1))); \\ Michel Marcus, Dec 12 2017

Crossrefs

Row sums give A003418.

Cf. A002088 (row lengths for n>=1).

Cf. A013595, A013596, A076585.

Sequence in context: A165621 A284321 A004739 * A203776 A343559 A242357

Adjacent sequences: A156279 A156280 A156281 * A156283 A156284 A156285

Keyword

nonn,tabf

Author

Roger L. Bagula, Feb 07 2009

Status

approved