referencesfor these coefficients



Here's a triangle which I find somewhat interesting. The first few rows are:


1
1 1
1 4 1
1 11 11 1
1 26 66 26 1
1 57 302 302 57 1
1 120 1191 2416 1191 120 1

The element at the intersection of the i_th and j_th diagonals is equal to
the linear combination of its predecessors on those diagonals, using the
weights i and j respectively.

(You might say that this is a souped-up version of Pascal's triangle.)

It is amusing that the sum of the elements of the n_th row is the factorial
of n.

The triangle is actually useful, it seems. For example, the series

1 + 8x + 27x^2 + 64x^3 + ... = (1 + 4x + x^2) / (1 - x)^4 for |x| <
1.

The coefficients of the series are the k_th powers of successive integers,
and the coefficients of the polynomial of degree k-1 in the numerator on the
RHS come from the triangle's k_th row; the exponent on (1-x) in the
denominator is k+1. (The power-series expansion of 1 / 1-x to a positive
integral power uses a diagonal of Pascal's triangle.)

I just stumbled on this today while experimenting. Where is it in the
literature? Are there any identities implied which are deeper than
superficial? (The part of about n!, for example.)




.

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