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A334369
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)*log(1 + x))/(1 - k*log(1 + x)).
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3
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1, 1, 1, 1, 1, -1, 1, 1, 1, 2, 1, 1, 3, 2, -6, 1, 1, 5, 14, 4, 24, 1, 1, 7, 38, 86, 14, -120, 1, 1, 9, 74, 384, 664, 38, 720, 1, 1, 11, 122, 1042, 4854, 6136, 216, -5040, 1, 1, 13, 182, 2204, 18344, 73614, 66240, 600, 40320, 1, 1, 15, 254, 4014, 49774, 387512, 1302552, 816672, 6240, -362880
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OFFSET
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0,10
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LINKS
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FORMULA
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T(0,k)=1 and T(n,k) = Sum_{j=0..n} j! * k^(j-1) * Stirling1(n,j) for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
-1, 1, 3, 5, 7, 9, ...
2, 2, 14, 38, 74, 122, ...
-6, 4, 86, 384, 1042, 2204, ...
24, 14, 664, 4854, 18344, 49774, ...
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MATHEMATICA
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T[0, k_] = 1; T[n_, k_] := Sum[If[k == 0 && j <= 1, 1, k^(j - 1)] * j! * StirlingS1[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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