A142724 Irregular triangle read by rows: row n gives coefficients in expansion of Product_{k=1..n} (1 + x^(2*k + 1)) for n >= 0.

1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2

Offset

0,43

Comments

For n >= 1, row n is the Poincaré polynomial for the Lie group A_n.

Row sums are powers of 2.

References

Borel, A. and Chevalley, C., The Betti numbers of the exceptional groups, Mem. Amer. Math. Soc. 1955, no. 14, pp 1-9.

Samuel I. Goldberg, Curvature and Homology, Dover, New York, 1998, page 144

Links

Alois P. Heinz, Rows n = 0..40, flattened

Formula

p(x,n) = Product[(1 + x^(2*k + 1)), {k, 1, n}]; t(n,m)=coefficients(p(x,n)).

Example

Triangle begins:
{1} (the empty product)
{1, 0, 0, 1},
{1, 0, 0, 1, 0, 1, 0, 0, 1},
{1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1},
{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1},
{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1},
...

Maple

A:=n->mul(1+x^(2*r+1), r=1..n);
for n from 1 to 12 do lprint(seriestolist(series(A(n), x, 10000))); od:

Mathematica

Clear[p, x, n, m]; p[x_, n_] = Product[(1 + x^(2*k + 1)), {k, 1, n}]; Table[CoefficientList[p[x, n], x], {n, 1, 10}]; Flatten[%]

Crossrefs

Cf. A178600, A178602, A178580, A178651, A178665, A178666.

Rows: A178686. A178687, A178691, A178692, A178695, A178696, A178697.

Sequence in context: A184159 A231122 A178686 * A174656 A364047 A178798

Adjacent sequences: A142721 A142722 A142723 * A142725 A142726 A142727

Keyword

nonn,tabf

Author

Roger L. Bagula and Gary W. Adamson, Sep 26 2008

Extensions

Edited by N. J. A. Sloane, Dec 25 2010

Status

approved