The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A246522 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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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.
(history; published version)

#24 by Michael De Vlieger at Wed Nov 22 08:16:44 EST 2023

STATUS

reviewed

approved

#23 by Stefano Spezia at Wed Nov 22 07:59:09 EST 2023

STATUS

proposed

reviewed

#22 by Jean-François Alcover at Wed Nov 22 05:47:13 EST 2023

STATUS

editing

proposed

#21 by Jean-François Alcover at Wed Nov 22 05:47:06 EST 2023

MATHEMATICA

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 *)

STATUS

approved

editing

#20 by Alois P. Heinz at Sun May 01 14:02:00 EDT 2022

STATUS

proposed

approved

#19 by Jean-François Alcover at Sun May 01 05:40:20 EDT 2022

STATUS

editing

proposed

#18 by Jean-François Alcover at Sun May 01 05:39:56 EDT 2022

MATHEMATICA

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}];

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 *)

STATUS

approved

editing

Discussion

Sun May 01

05:40

Jean-François Alcover: Minor edit of Mma code

#17 by Alois P. Heinz at Sun Feb 24 11:16:33 EST 2019

STATUS

editing

approved

#16 by Alois P. Heinz at Sun Feb 24 11:02:13 EST 2019

EXAMPLE

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, ...

STATUS

approved

editing

#15 by Alois P. Heinz at Thu Dec 04 07:26:06 EST 2014

STATUS

proposed

approved