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A089944 revision #4

A089944

Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k)=(k+1) for k>=0.

1, 2, 1, 3, 3, 1, 4, 8, 4, 1, 5, 20, 15, 5, 1, 6, 48, 54, 24, 6, 1, 7, 112, 189, 112, 35, 7, 1, 8, 256, 648, 512, 200, 48, 8, 1, 9, 576, 2187, 2304, 1125, 324, 63, 9, 1, 10, 1280, 7290, 10240, 6250, 2160, 490, 80, 10, 1, 11, 2816, 24057, 45056, 34375, 14256, 3773, 704

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The main diagonal is A089945: {T(n,n)=(2*n+1)*(n+1)^(n-1), n>=0}; the hyperbinomial transform of the main diagonal is the next lower diagonal in the array (A089946): {T(n+1,n)=2*(n+1)*(n+2)^(n-1), n>=0}.

LINKS

Table of n, a(n) for n=0..62.

FORMULA

T(n, k)=(k+n+1)*(n+1)^(k-1). E.g.f.: (1+x)*exp(x)/((1-y*exp(x)).

EXAMPLE

Rows begin:

{1, 2, 3, 4, 5, 6, 7,..},

{1, 3, 8, 20, 48, 112, 256,..},

{1, 4, 15, 54, 189, 648, 2187,..},

{1, 5, 24, 112, 512, 2304, 10240,..},

{1, 6, 35, 200, 1125, 6250, 34375,..},

{1, 7, 48, 324, 2160, 14256, 93312,..},