%I #45 Jul 15 2022 08:10:11
%S 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,
%T 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,
%U 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
%N Irregular triangle read by rows: row n is the expansion of the product of cyclotomic polynomials from 2nd to n-th polynomial.
%H Alois P. Heinz, <a href="/A156282/b156282.txt">Rows n = 0..60, flattened</a>
%H L. Carlitz, <a href="https://projecteuclid.org/euclid.dmj/1077475200">q-Bernoulli numbers and polynomials</a>, Duke Math. J. Volume 15, Number 4 (1948), 987-1000.
%H L. Carlitz and J. Riordan, <a href="https://projecteuclid.org/euclid.dmj/1077375351">Two element lattice permutation numbers and their q-generalization</a>, Duke Math. J. Volume 31, Number 3 (1964), 371-388
%H Y.-H. He, C. Matti, and C. Sun, <a href="http://arxiv.org/abs/1404.6833">The Scattering Variety</a>, arXiv preprint arXiv:1403.6833 [cs.SE], 2014. See Table 2, central column. - _N. J. A. Sloane_, Jun 28 2014
%H John Shareshian and Michelle L. Wachs, <a href="http://arxiv.org/abs/math/0608274">q-Eulerian Polynomials: Excedance Number and Major Index</a>, arXiv:math/0608274 [math.CO], 2006, page 3.
%e 1;
%e 1;
%e 1, 1;
%e 1, 2, 2, 1;
%e 1, 2, 3, 3, 2, 1;
%e 1, 3, 6, 9, 11, 11, 9, 6, 3, 1;
%e 1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1;
%e 1, 3, 7, 13, 21, 30, 39, 46, 50, 50, 46, 39, 30, 21, 13, 7, 3, 1;
%e 1, 3, 7, 13, 22, 33, 46, 59, 71, 80, 85, 85, 80, 71, 59, 46, 33, 22, 13, 7, 3, 1;
%e 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
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
%p mul(numtheory[cyclotomic](i, x), i=2..n)):
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Jul 15 2022
%t row[n_] := CoefficientList[Product[Cyclotomic[k+1, x], {k, 1, n}], x];
%t Table[row[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jul 15 2022 *)
%o (PARI) row(n) = Vec(prod(k=1, n, polcyclo(k+1))); \\ _Michel Marcus_, Dec 12 2017
%Y Row sums give A003418.
%Y Cf. A002088 (row lengths for n>=1).
%Y Cf. A013595, A013596, A076585.
%K nonn,tabf
%O 0,6
%A _Roger L. Bagula_, Feb 07 2009