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A292977 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).

%I A292977 #10 Mar 08 2018 13:19:45
%S A292977 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 A292977 720,1,-5,17,-34,33,8,265,5040,1,-6,26,-74,120,-78,112,1854,40320,1,
%U A292977 -7,37,-138,329,-424,261,656,14833,362880,1,-8,50,-232,744,-1480,1552,-360,5504,133496,3628800
%N A292977 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 A292977 A(n,k) is the k-th inverse binomial transform of A000142 evaluated at n.
%C A292977 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 A292977 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A292977 T(n, k) = n! * Sum_{j=0..n} (-k)^j/j!. - _Max Alekseyev_, Mar 06 2018
%F A292977 E.g.f. of column k: exp(-k*x)/(1 - x).
%e A292977 Square array begins:
%e A292977 n=0:    1,   1,   1,    1,     1,      1,  ...
%e A292977 n=1:    1,   0,  -1,   -2,    -3,     -4,  ...
%e A292977 n=2:    2,   1,   2,    5,    10,     17,  ...
%e A292977 n=3:    6,   2,  -2,  -12,   -34,    -74,  ...
%e A292977 n=4:   24,   9,   8,   33,   120,    329,  ...
%e A292977 n=5:  120,  44,   8,  -78,  -424,  -1480,  ...
%e A292977 ...
%e A292977 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 A292977 Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t A292977 FullSimplify[Table[Function[k, Exp[-k] Gamma[n + 1, -k]][j - n], {j, 0, 10}, {n, 0, j}]] // Flatten
%Y A292977 Columns: A000142 (k=0), A000166 (k=1), A000023 (k=2), A010843 (k=3, with offset 0).
%Y A292977 Main diagonal: A134095 (absolute values).
%Y A292977 Cf. A080955, A089258.
%K A292977 sign,tabl
%O A292977 0,6
%A A292977 _Ilya Gutkovskiy_, Sep 27 2017