A170956 -id:A170956 - cp's OEIS frontend

A178725 Irregular triangle read by rows: row n gives coefficients in expansion of Product_{k=1..n} (1 + x^(4*k - 1)) for n >= 0.

1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 2, 1, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 1, 0, 1, 3, 0, 0, 2, 2, 0, 0, 3, 2, 0, 1, 3, 1, 0, 1, 3, 1, 0, 2, 3, 0, 0, 2, 2, 0, 0, 3, 1, 0, 1, 2, 1, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1

Offset: 0

N. J. A. Sloane, Dec 26 2010

For n >= 1, row n is the Poincaré polynomial for the Lie group B_n (or, equally, Sp(2n) or O(2n+1)).

Row sums are powers of 2.

			Triangle begins:
[1] (the empty product)
[1, 0, 0, 1]
[1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1]
[1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1]
[1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1]
[1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 2, 1, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1]
...

Borel, A. and Chevalley, C., The Betti numbers of the exceptional groups, Mem. Amer. Math. Soc. 1955, no. 14, pp 1-9.
H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Rows: A170956-A170965. Cf. A142724.

cp's OEIS Frontend

A178725 Irregular triangle read by rows: row n gives coefficients in expansion of Product_{k=1..n} (1 + x^(4*k - 1)) for n >= 0.

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