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A246522
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Number A(n,k) of endofunctions on [n] whose cycle lengths are divisors of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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11
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1, 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 1, 4, 16, 0, 1, 1, 3, 25, 125, 0, 1, 1, 4, 18, 218, 1296, 0, 1, 1, 3, 25, 157, 2451, 16807, 0, 1, 1, 4, 16, 224, 1776, 33832, 262144, 0, 1, 1, 3, 27, 125, 2601, 24687, 554527, 4782969, 0, 1, 1, 4, 16, 250, 1320, 37072, 407464, 10535100, 100000000, 0
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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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E.g.f. of column k: exp(Sum_{d|k} (-LambertW(-x))^d/d).
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, ...
0, 3, 4, 3, 4, 3, 4, ...
0, 16, 25, 18, 25, 16, 27, ...
0, 125, 218, 157, 224, 125, 250, ...
0, 1296, 2451, 1776, 2601, 1320, 2951, ...
0, 16807, 33832, 24687, 37072, 17671, 42552, ...
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MAPLE
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with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
A:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1, k)*
(i-1)!^j, j=0..`if`(irem(k, i)=0, n/i, 0))))
end:
A:=(n, k)->add(b(j, min(k, j), k)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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egf[k_] := Exp[Sum[(-ProductLog[-x])^d/d, {d, Divisors[k]}]];
A[1, 0] = 0; A[0, _] = 1; A[1, _] = 1; A[_, 0] = 0;
A[n_, k_] := n!*SeriesCoefficient[egf[k], {x, 0, n}];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 04 2014, translated from first Maple program *)
multinomial[n_, k_List] := n!/Times @@ (k!);
Unprotect[Power]; 0^0 = 1; Protect[Power];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, {j}]]]/j!*b[n - i*j, i-1, k]*(i-1)!^j, {j, 0, If[Mod[k, i] == 0, n/i, 0]}]]];
A[n_, k_] := Sum[b[j, Min[k, j], k]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 22 2023, from 2nd Maple program *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000272(n+1), A209319, A246523, A246524, A246525, A246526, A246527, A246528, A246529, A246530.
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KEYWORD
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AUTHOR
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STATUS
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approved
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