A208248 -id:A208248

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A290932

Sum of the LCM of cycle lengths over all endofunctions on [n].

+10
5

1, 1, 5, 40, 431, 5886, 96817, 1862890, 41043375, 1018584610, 28108489541, 853617865134, 28287119604955, 1015630741097350, 39273014068691145, 1627118268024495586, 71904849762914854703, 3375959341815207350850, 167810405947367539063885, 8803814897608815310714270

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..150

FORMULA

a(n) = Sum_{k=1..A000793(n)} k * A222029(n,k).

MAPLE

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*

b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n))

end:

a:= n-> add(b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n):

seq(a(n), n=0..25);

MATHEMATICA

T[n_, k_] := T[n, k] = If[n == 0, k, Sum[(j - 1)! * T[n - j, LCM[k, j]]*Binomial[n - 1, j - 1], {j, n}]]; {1}~Join~Table[Sum[T[j, 1]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}], {n, 19}] (* Michael De Vlieger, Aug 17 2017 *)

CROSSREFS

Cf. A000793, A060014, A208231, A208248, A222029.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 13 2017

STATUS

approved

A208231

Sum of the minimum cycle length over all functions f:{1,2,...,n}->{1,2,...,n} (endofunctions).

+10
3

0, 1, 5, 37, 373, 4761, 73601, 1336609, 27888281, 657386305, 17276807089, 500876786301, 15879053677697, 546470462226313, 20288935994319929, 808320431258439121, 34397370632215764001, 1557106493482564625793, 74713970491718324746529, 3787792171563440619543133, 202314171910557294992453009

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Sum of the number of endofunctions whose cycle lengths are >=i for all i >=1. A000312 + A065440 + A134362 + A208230 + ...

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..386

FORMULA

E.g.f.: A(T(x)) = Sum_{k>=1} exp( Sum_{i>=k} T(x)^i/i) - 1 where A(x) is the e.g.f. for A028417 and T(x) is the e.g.f. for A000169.

MAPLE

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*

b(n-j, min(m, j))*binomial(n-1, j-1), j=1..n))

end:

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

seq(a(n), n=0..25); # Alois P. Heinz, May 20 2016

MATHEMATICA

nn=20; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Apply[Plus, Table[Range[0, nn]!CoefficientList[Series[Exp[Sum[t^i/i, {i, n, nn}]]-1, {x, 0, nn}], x], {n, 1, nn}]]