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A246609 Number T(n,k) of endofunctions on [n] whose cycle lengths are multiples of k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 11
1, 0, 1, 0, 4, 1, 0, 27, 6, 2, 0, 256, 57, 24, 6, 0, 3125, 680, 300, 120, 24, 0, 46656, 9945, 4480, 2160, 720, 120, 0, 823543, 172032, 78750, 41160, 17640, 5040, 720, 0, 16777216, 3438673, 1591296, 866460, 430080, 161280, 40320, 5040 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. T(0,k) = 1, T(n,k) = 0 for k>n and n>0.

Column k > 1 is asymptotic to n^(n - 1/2 + 1/(2*k)) * sqrt(2*Pi) / (2^(1/(2*k)) * k^(1/k) * GAMMA(1/(2*k))) * (1 - (3*k-1)*(k-1) * sqrt(2/n) * GAMMA(1/(2*k)) / (12 * k^2 * GAMMA(1/2+1/(2*k)))). - Vaclav Kotesovec, Sep 01 2014

LINKS

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

FORMULA

E.g.f. for column k > 0: 1 / (1 - (-1)^k * LambertW(-x)^k)^(1/k). - Vaclav Kotesovec, Sep 01 2014

EXAMPLE

Triangle T(n,k) begins:

1;

0,      1;

0,      4,      1;

0,     27,      6,     2;

0,    256,     57,    24,     6;

0,   3125,    680,   300,   120,    24;

0,  46656,   9945,  4480,  2160,   720,  120;

0, 823543, 172032, 78750, 41160, 17640, 5040, 720;

MAPLE

with(combinat):

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

      `if`(i=0 or i>n, 0, add(b(n-i*j, i+k, k)*(i-1)!^j*

      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))

    end:

T:= (n, k)->add(b(j, k$2)*n^(n-j)*binomial(n-1, j-1), j=0..n):

seq(seq(T(n, k), k=0..n), n=0..10);

MATHEMATICA

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0 || i > n, 0, Sum[b[n-i*j, i+k, k]*(i-1)!^j*multinomial[n, {n-i*j, Sequence @@ Table[i, {j}]}]/j!, {j, 0, n/i}]]]; T[0, 0] = 1; T[n_, k_] := Sum[b[j, k, k]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-Fran├žois Alcover, Jan 06 2015, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000007, A000312, A060435, A246610, A246611, A246612, A246613, A246614, A246615, A246616, A246617.

Main diagonal gives A000142(n-1) for n>0.

T(2n,n) gives A246618.

Sequence in context: A081114 A069018 A156811 * A130636 A299354 A117414

Adjacent sequences:  A246606 A246607 A246608 * A246610 A246611 A246612

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 31 2014

STATUS

approved

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Last modified August 3 05:55 EDT 2020. Contains 336197 sequences. (Running on oeis4.)