A246523 -id:A246523 - cp's OEIS frontend

A246522 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.

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

Offset: 0

Alois P. Heinz, Aug 28 2014

			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, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000272(n+1), A209319, A246523, A246524, A246525, A246526, A246527, A246528, A246529, A246530.

Main diagonal gives A246531.

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);

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 *)

E.g.f. of column k: exp(Sum_{d|k} (-LambertW(-x))^d/d).

A246529 Number of endofunctions on [n] whose cycle lengths are divisors of 9.

Original entry on oeis.org

1, 1, 3, 18, 157, 1776, 24687, 407464, 7792857, 169594560, 4141165051, 112178655744, 3339749183157, 108422228887168, 3812520677598375, 144372964560581376, 5858088633723823153, 253575577033176047616, 11664031615012086920307, 568166632439929892761600

Offset: 0

Alois P. Heinz, Aug 28 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..350

Column k=9 of A246522.

with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(9), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) 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)*
      (i-1)!^j, j=0..`if`(irem(9, i)=0, n/i, 0))))
    end:
a:= n-> add(b(j, min(9, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{d|9} (-LambertW(-x))^d/d).

a(n) - A246523(n) is a multiple of 40320. - M. F. Hasler, Oct 26 2014

cp's OEIS Frontend

A246522 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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