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%I #9 Feb 08 2018 14:16:25
%S 1,1,1,1,0,2,1,-1,-1,6,1,-2,-2,-4,24,1,-3,-1,-2,-15,120,1,-4,2,6,8,
%T -56,720,1,-5,7,14,33,88,-185,5040,1,-6,14,16,24,102,592,-204,40320,1,
%U -7,23,6,-31,-104,-9,3344,6209,362880,1,-8,34,-22,-120,-380,-1328,-3762,14464,112400,3628800
%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))/(1 - x).
%H G. C. Greubel, <a href="/A295381/b295381.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F E.g.f. of column k: exp(-k*x/(1 - x))/(1 - x).
%F A(n,k) = n!*Laguerre(n,k).
%e E.g.f. of column k: A_k(x) = 1 + (1 - k)*x/1! + (k^2 - 4*k + 2)*x^2/2! + (-k^3 + 9*k^2 - 18*k + 6)*x^3/3! + (k^4 - 16*k^3 + 72*k^2 - 96*k + 24)*x^4/4! + ...
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 0, -1, -2 -3, -4, ...
%e 2, -1, -2, -1, 2, 7, ...
%e 6, -4, -2, 6, 14, 16, ...
%e 24, -15, 8, 33, 24, -31, ...
%e 120, -56, 88, 102, -104, -380, ...
%t Table[Function[k, n! SeriesCoefficient[Exp[-k x/(1 - x)]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t Table[Function[k, n! LaguerreL[n, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t Table[Function[k, n! Hypergeometric1F1[-n, 1, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%Y Columns k=0..2 give A000142, A009940, A295382.
%Y Main diagonal gives A277423.
%Y Cf. A289192.
%K sign,tabl
%O 0,6
%A _Ilya Gutkovskiy_, Nov 21 2017
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