A142724 - OEIS

%I #24 Aug 21 2015 14:29:10

%S 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,

%T 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,

%U 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

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

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

%C Row sums are powers of 2.

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

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

%H Alois P. Heinz, <a href="/A142724/b142724.txt">Rows n = 0..40, flattened</a>

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

%e Triangle begins:

%e {1} (the empty product)

%e {1, 0, 0, 1},

%e {1, 0, 0, 1, 0, 1, 0, 0, 1},

%e {1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1},

%e {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},

%e {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},

%e ...

%p A:=n->mul(1+x^(2*r+1),r=1..n);

%p for n from 1 to 12 do lprint(seriestolist(series(A(n),x,10000))); od:

%t 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[%]

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

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

%K nonn,tabf

%O 0,43

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 26 2008

%E Edited by _N. J. A. Sloane_, Dec 25 2010