Mistakes in A035607 ?
Roger Cuculière
rcuculiere at free.fr
Mon Apr 10 12:03:53 CEST 2006
The sequence A035607 is the triangle of a(n,p), the number of n-uples
(x_1, x_2,...,x_n) of rational integers such that :
|x_1| + |x_2| + ... + |x_n| = p, for n>=1 and p>=0.
The generating function G(x,y) of this double sequence is the sum of
a(n,p)*x^n*y^p, n=1..infty, p=0..infty. I think that this function is :
G(x,y)=x*(1+y)/(1-x-y-(x*y)), and not (1+x)/(1-x-x*y-x^2*y), as claimed
in the notice.
Furthermore, the horizontal generating function H_n(y), which generates
the rows of the table : (1, 2, 2, 2, 2,...), (1, 4, 8, 12, 16, ...), (1,
6, 18, 38, 66, ...), is the sum of a(n,p)*y^p, p=0..infty, for each
fixed n. I think it is : H_n(y)=((1+y)^n)/((1-y)^n)) an not :
"(1+n)^k/(1-n)^(-k)", very strange function.
And the vertical generating function V_p(x), which generates the rows of
the table : (1, 1, 1, 1, 1, ...}, (2, 4, 6, 8, 10, ...), (2, 8, 18, 32,
50, ...), is the sum of a(n,p)*x^n, n=1..infty, for each fixed p. I
think it is : V_p(x)=2*((1+x)^(p-1))/((1-x)^(p+1)) for p>=1, and :
V_0(x)=x/(1-x).
Best regards from Clichy,
Roger CUCULIÈRE
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