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A292977
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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).
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3
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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, 720, 1, -5, 17, -34, 33, 8, 265, 5040, 1, -6, 26, -74, 120, -78, 112, 1854, 40320, 1, -7, 37, -138, 329, -424, 261, 656, 14833, 362880, 1, -8, 50, -232, 744, -1480, 1552, -360, 5504, 133496, 3628800
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OFFSET
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0,6
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COMMENTS
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A(n,k) is the k-th inverse binomial transform of A000142 evaluated at n.
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
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LINKS
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FORMULA
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T(n, k) = n! * Sum_{j=0..n} (-k)^j/j!. - Max Alekseyev, Mar 06 2018
E.g.f. of column k: exp(-k*x)/(1 - x).
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EXAMPLE
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Square array begins:
n=0: 1, 1, 1, 1, 1, 1, ...
n=1: 1, 0, -1, -2, -3, -4, ...
n=2: 2, 1, 2, 5, 10, 17, ...
n=3: 6, 2, -2, -12, -34, -74, ...
n=4: 24, 9, 8, 33, 120, 329, ...
n=5: 120, 44, 8, -78, -424, -1480, ...
...
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! + ...
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MATHEMATICA
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Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
FullSimplify[Table[Function[k, Exp[-k] Gamma[n + 1, -k]][j - n], {j, 0, 10}, {n, 0, j}]] // Flatten
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CROSSREFS
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Main diagonal: A134095 (absolute values).
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KEYWORD
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AUTHOR
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STATUS
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approved
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