login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 14
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 (list; graph; refs; listen; history; text; internal format)
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 A178798 A318277

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 16 07:30 EDT 2019. Contains 328051 sequences. (Running on oeis4.)