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%I #10 Mar 08 2018 13:19:45
%S 1,1,1,1,0,2,1,-1,1,6,1,-2,2,2,24,1,-3,5,-2,9,120,1,-4,10,-12,8,44,
%T 720,1,-5,17,-34,33,8,265,5040,1,-6,26,-74,120,-78,112,1854,40320,1,
%U -7,37,-138,329,-424,261,656,14833,362880,1,-8,50,-232,744,-1480,1552,-360,5504,133496,3628800
%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x).
%C A(n,k) is the k-th inverse binomial transform of A000142 evaluated at n.
%C Can be considered as extension of the array A089258 to columns with negative indices via A089258(n,k) = A(n,-k) or, vice versa, A(n,k) = A089258(n,-k). - _Max Alekseyev_, Mar 06 2018
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F T(n, k) = n! * Sum_{j=0..n} (-k)^j/j!. - _Max Alekseyev_, Mar 06 2018
%F E.g.f. of column k: exp(-k*x)/(1 - x).
%e Square array begins:
%e n=0: 1, 1, 1, 1, 1, 1, ...
%e n=1: 1, 0, -1, -2, -3, -4, ...
%e n=2: 2, 1, 2, 5, 10, 17, ...
%e n=3: 6, 2, -2, -12, -34, -74, ...
%e n=4: 24, 9, 8, 33, 120, 329, ...
%e n=5: 120, 44, 8, -78, -424, -1480, ...
%e ...
%e E.g.f. of column k: A_k(x) = 1 + (1 - k)*x/1! + (k^2 - 2*k + 2)*x^2/2! + (-k^3 + 3*k^2 - 6*k + 6) x^3/3! + (k^4 - 4*k^3 + 12*k^2 - 24*k + 24)*x^4/4! + ...
%t Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t FullSimplify[Table[Function[k, Exp[-k] Gamma[n + 1, -k]][j - n], {j, 0, 10}, {n, 0, j}]] // Flatten
%Y Columns: A000142 (k=0), A000166 (k=1), A000023 (k=2), A010843 (k=3, with offset 0).
%Y Main diagonal: A134095 (absolute values).
%Y Cf. A080955, A089258.
%K sign,tabl
%O 0,6
%A _Ilya Gutkovskiy_, Sep 27 2017