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Square array of numbers related to the incomplete gamma function, read by antidiagonals.
7

%I #16 Mar 08 2018 13:18:53

%S 1,1,1,1,2,2,1,3,5,6,1,4,10,16,24,1,5,17,38,65,120,1,6,26,78,168,326,

%T 720,1,7,37,142,393,872,1957,5040,1,8,50,236,824,2208,5296,13700,

%U 40320,1,9,65,366,1569,5144,13977,37200,109601,362880,1,10,82,538,2760,10970,34960,100026

%N Square array of numbers related to the incomplete gamma function, read by antidiagonals.

%F T(k,n) = n! * Sum{j=0..n} k^j/j!.

%F E.g.f. of k-th row: exp(k*x)/(1-x).

%F T(k,n) = A089258(n,k).

%e Array begins:

%e k=0: 1 1 2 6 24 ...

%e k=1: 1 2 5 16 65 ...

%e k=2: 1 3 10 38 168 ...

%e k=3: 1 4 17 78 393 ...

%e k=4: 1 5 26 142 824 ...

%e ...

%t T[0, k_] := k!; T[n_, k_] := k!*Sum[n^j/j!, {j, 0, k}];

%t Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 17 2018 *)

%Y Rows: A000142, A000522, A010842, A053486, A053487, A080954.

%Y Transposed version: A089258.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Feb 26 2003

%E Corrected by _Philippe Deléham_, Dec 12 2003