Search: a245507 -id:a245507
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A245501
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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.
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+10
11
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1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 27, 1, 1, 1, 4, 10, 256, 1, 1, 1, 3, 19, 41, 3125, 1, 1, 1, 4, 12, 110, 196, 46656, 1, 1, 1, 3, 19, 73, 751, 1057, 823543, 1, 1, 1, 4, 10, 116, 556, 5902, 6322, 16777216, 1, 1, 1, 3, 21, 41, 901, 4737, 52165, 41393, 387420489, 1
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OFFSET
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0,9
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LINKS
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FORMULA
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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.
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EXAMPLE
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Square array A(n,k) begins:
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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MAPLE
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with(numtheory):
A:= (n, k)-> `if`(k=0, 1, `if`(k=1, n^n, n! *coeff(series(
exp(add((x*exp(x))^d/d, d=divisors(k-1))), x, n+1), x, n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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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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CROSSREFS
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Column k=0-10 give: A000012, A000312, A000248, A060905, A060906, A060907, A245502, A245503, A245504, A245505, A245506.
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KEYWORD
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AUTHOR
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STATUS
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approved
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