%I #24 Nov 22 2023 08:16:44

%S 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,

%T 1,3,25,157,2451,16807,0,1,1,4,16,224,1776,33832,262144,0,1,1,3,27,

%U 125,2601,24687,554527,4782969,0,1,1,4,16,250,1320,37072,407464,10535100,100000000,0

%N 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.

%H Alois P. Heinz, <a href="/A246522/b246522.txt">Antidiagonals n = 0..140, flattened</a>

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

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 1, 1, 1, 1, 1, ...

%e 0, 3, 4, 3, 4, 3, 4, ...

%e 0, 16, 25, 18, 25, 16, 27, ...

%e 0, 125, 218, 157, 224, 125, 250, ...

%e 0, 1296, 2451, 1776, 2601, 1320, 2951, ...

%e 0, 16807, 33832, 24687, 37072, 17671, 42552, ...

%p with(numtheory):

%p egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):

%p A:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

%p # second Maple program:

%p with(combinat):

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1, k)*

%p (i-1)!^j, j=0..`if`(irem(k, i)=0, n/i, 0))))

%p end:

%p A:=(n, k)->add(b(j, min(k, j), k)*n^(n-j)*binomial(n-1, j-1), j=0..n):

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

%t egf[k_] := Exp[Sum[(-ProductLog[-x])^d/d, {d, Divisors[k]}]];

%t A[1, 0] = 0; A[0, _] = 1; A[1, _] = 1; A[_, 0] = 0;

%t A[n_, k_] := n!*SeriesCoefficient[egf[k], {x, 0, n}];

%t Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 04 2014, translated from first Maple program *)

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t Unprotect[Power]; 0^0 = 1; Protect[Power];

%t 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]}]]];

%t A[n_, k_] := Sum[b[j, Min[k, j], k]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}];

%t Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 22 2023, from 2nd Maple program *)

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

%Y Main diagonal gives A246531.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Aug 28 2014