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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. 2
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,..},

{1, 8, 63, 490, 3773, 28812, 218491,..},..

PROG

(PARI) T(n, k)=if(n<0 || k<0, 0, (k+n+1)*(n+1)^(k-1))

CROSSREFS

Cf. A089945, A089946.

Rows : A000027, A001792, A006234, A079028, A081105, A081106, A081107, A081108, A081109, A081122.

Columns : A000012, A000027, A005563.

Sequence in context: A208335 A208597 A179943 * A180165 A142249 A274705

Adjacent sequences:  A089941 A089942 A089943 * A089945 A089946 A089947

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Nov 23 2003

STATUS

approved

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Last modified November 29 10:41 EST 2021. Contains 349416 sequences. (Running on oeis4.)