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A284816 Sum of entries in the first cycles of all permutations of [n]. 5
1, 4, 21, 132, 960, 7920, 73080, 745920, 8346240, 101606400, 1337212800, 18920563200, 286442956800, 4620449433600, 79114299264000, 1433211107328000, 27387931963392000, 550604138692608000, 11617107089043456000, 256671161862635520000, 5926549291918295040000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..448

Wikipedia, Permutation

FORMULA

a(n) = n!*(n*(n+1)-(n-1)*(n+2)/2)/2.

E.g.f.: -x*(x^2-2*x+2)/(2*(x-1)^3).

a(n) = (n^2+n+2)*n*a(n-1)/(n^2-n+2) for n>1, a(n) = n for n<2.

EXAMPLE

a(3) = 21 because the sum of the entries in the first cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 6+6+3+4+1+1 = 21.

MAPLE

a:= n-> n!*(n*(n+1)-(n-1)*(n+2)/2)/2:

seq(a(n), n=1..25);

# second Maple program:

a:= proc(n) option remember; `if`(n<2, n,

       (n^2+n+2)*n*a(n-1)/(n^2-n+2))

    end:

seq(a(n), n=1..25);

CROSSREFS

Cf. A180119, A185105, A285363, A285382.

Column k=1 of A285439.

Sequence in context: A131965 A332851 A303563 * A226067 A104982 A306335

Adjacent sequences:  A284813 A284814 A284815 * A284817 A284818 A284819

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Apr 15 2017

STATUS

approved

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Last modified May 10 22:16 EDT 2021. Contains 343780 sequences. (Running on oeis4.)