A289957
Number of permutations of [n] having exactly nine nontrivial cycles.
Original entry on oeis.org
34459425, 4583103525, 353553700500, 20941727706900, 1063956789420525, 49122916648430625, 2133284602385096160, 89095866141017736000, 3632887356673057858375, 146162134100332940264675, 5846991791902793693661500, 233881907399101905866222300
Offset: 18
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Drop[CoefficientList[Series[(-Log[1 - x] - x)^9/9!*Exp[x] , {x, 0, 50}], x] * Table[k !, {k, 0, 50}] , 18] (* Indranil Ghosh, Jul 16 2017 *)
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x = 'x + O('x^30); Vec(serlaplace((-log(1-x)-x)^9/9!*exp(x))) \\ Michel Marcus, Jul 16 2017
A289958
Number of permutations of [n] having exactly ten nontrivial cycles.
Original entry on oeis.org
654729075, 105411381075, 9729928783575, 682187321190375, 40619713503218850, 2178070341434237250, 108946451099337101250, 5201416870471272701250, 240786076823686096077375, 10929696310323023478019775, 490470855046984824670532875, 21893162839119896036432594875
Offset: 20
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Drop[CoefficientList[Series[(Log[1 - x] + x)^10/10!*Exp[x] , {x, 0, 50}], x] * Table[k!, {k, 0, 50}] , 20] (* Indranil Ghosh, Jul 16 2017 *)
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x = 'x + O('x^30); Vec(serlaplace((-log(1-x)-x)^10/10!*exp(x))) \\ Michel Marcus, Jul 16 2017
A350446
Number T(n,k) of endofunctions on [n] with exactly k cycles of length larger than 1; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
Original entry on oeis.org
1, 1, 3, 1, 16, 11, 125, 128, 3, 1296, 1734, 95, 16807, 27409, 2425, 15, 262144, 499400, 61054, 945, 4782969, 10346328, 1605534, 42280, 105, 100000000, 240722160, 44981292, 1706012, 11025, 2357947691, 6222652233, 1351343346, 67291910, 763875, 945
Offset: 0
Triangle T(n,k) begins:
1;
1;
3, 1;
16, 11;
125, 128, 3;
1296, 1734, 95;
16807, 27409, 2425, 15;
262144, 499400, 61054, 945;
4782969, 10346328, 1605534, 42280, 105;
100000000, 240722160, 44981292, 1706012, 11025;
2357947691, 6222652233, 1351343346, 67291910, 763875, 945;
...
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c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
t:= proc(n) option remember; n^(n-1) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
b(n-i)*binomial(n-1, i-1)*(c(i)*x+t(i)), i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);
# second Maple program:
egf := k-> (LambertW(-x)-log(1+LambertW(-x)))^k/(exp(LambertW(-x))*k!):
A350446 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A350446(n, k), k=0..n/2)), n=0..10); # Mélika Tebni, Mar 23 2023
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c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
t[n_] := t[n] = n^(n - 1);
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
b[n - i]*Binomial[n - 1, i - 1]*(c[i]*x + t[i]), {i, 1, n}]]];
T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)
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