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Revision History for A245501
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
Number A(n,k) of endofunctions f on [n] such that f^k(i) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
(history; published version)
(history; published version)
EXAMPLE
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 4, 3, 4, 3, 4, 3, ...
1, 27, 10, 19, 12, 19, 10, ...
1, 256, 41, 110, 73, 116, 41, ...
1, 3125, 196, 751, 556, 901, 220, ...
1, 46656, 1057, 5902, 4737, 8422, 1921, ...
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MATHEMATICA
A[0, 1] = 1; A[n_, k_] := If[k==0, 1, If[k==1, n^n, n!*SeriesCoefficient[ Exp[ DivisorSum[k-1, (x*Exp[x])^#/#&]], {x, 0, n}]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 20 2017, translated from Maple *)
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LINKS
Alois P. Heinz, <a href="/A245501/b245501.txt">Rows Antidiagonals n = 0..140, flattened</a>
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FORMULA
A(n,k) = n! * [x^n] exp(Sum_{d|(k-1)} (x*exp(x))^d/d) for k>1, A(n,0)=1, A(n,1)=n^n.
A(n,1)=n^n.
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