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 A243098 Number T(n,k) of endofunctions on [n] with all cycles of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 8
 1, 0, 1, 0, 3, 1, 0, 16, 6, 2, 0, 125, 51, 24, 6, 0, 1296, 560, 300, 120, 24, 0, 16807, 7575, 4360, 2160, 720, 120, 0, 262144, 122052, 73710, 41160, 17640, 5040, 720, 0, 4782969, 2285353, 1430016, 861420, 430080, 161280, 40320, 5040 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(0,0) = 1 by convention. LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA E.g.f. of column k>0: exp((-LambertW(-x))^k/k), e.g.f. of column k=0: 1. EXAMPLE Triangle T(n,k) begins:   1;   0,      1;   0,      3,      1;   0,     16,      6,     2;   0,    125,     51,    24,     6;   0,   1296,    560,   300,   120,    24;   0,  16807,   7575,  4360,  2160,   720,  120;   0, 262144, 122052, 73710, 41160, 17640, 5040, 720; MAPLE with(combinat): T:= (n, k)-> `if`(k*n=0, `if`(k+n=0, 1, 0),     add(binomial(n-1, j*k-1)*n^(n-j*k)*(k-1)!^j*     multinomial(j*k, k\$j, 0)/j!, j=0..n/k)): seq(seq(T(n, k), k=0..n), n=0..10); MATHEMATICA multinomial[n_, k_] := n!/Times @@ (k!); T[n_, k_] := If[k*n==0, If[k+n == 0, 1, 0], Sum[Binomial[n-1, j*k-1]*n^(n-j*k)*(k-1)!^j*multinomial[j*k, Append[Array[k&, j], 0]]/j!, {j, 0, n/k}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *) CROSSREFS Columns k=0-4 give: A000007, A000272(n+1) for n>0, A057817(n+1), 2*A060917, 6*A060918. Row sums give A241980. T(2n,n) gives A246050. Main diagonal gives A000142(n-1) for n>0. Cf. A241981, A246049. Sequence in context: A131222 A228334 A114151 * A241981 A147723 A110518 Adjacent sequences:  A243095 A243096 A243097 * A243099 A243100 A243101 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Aug 18 2014 STATUS approved

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Last modified January 17 10:38 EST 2021. Contains 340214 sequences. (Running on oeis4.)