|
|
A151881
|
|
Sum (number of cycles)^2 over all n! permutations of [1..n].
|
|
7
|
|
|
1, 5, 23, 120, 724, 5012, 39332, 345832, 3371976, 36135792, 422379792, 5349561984, 72996193152, 1067779243008, 16670798231040, 276718772067840, 4866610479828480, 90401487246167040, 1768784607499944960, 36360467544043008000, 783508616506603008000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Sum (number of cycles) over all n! permutations of [1..n] gives A000254.
a(n) equals -1 times the coefficient of x of the characteristic polynomial of the n X n matrix whose (i,j)-entry is equal to i+1 if i=j, and is equal to 1 otherwise. [John M. Campbell, May 24 2011]
|
|
LINKS
|
|
|
FORMULA
|
a(n) = (-1)^(n+1)*(Stirling1(n+1,2)-2*Stirling1(n+1,3)). - Vladeta Jovovic, Jul 22 2009
a(n) = n!*(H(n)+H(n)^2-H2(n)), where H(n)=sum(k=1..n,1/k), H2(n)=sum(k=1..n,1/k^2). - Vladimir Reshetnikov, Oct 20 2015
a(n) = Sum_{k=0..n} |Stirling1(n,k)|*k^2. - Peter Luschny, Jan 14 2020
|
|
MAPLE
|
with(combinat): with(numtheory):
M:=30;
for n from 1 to M do
p:=partition(n); s:=0:
for k from 1 to nops(p) do
# get next partition of n
# convert partition to list of sizes of parts
q:=convert(p[k], multiset);
for i from 1 to n do a(i):=0: od:
for i from 1 to nops(q) do a(q[i][1]):=q[i][2]: od:
# get number of parts:
nump := add(a(i), i=1..n);
# get multiplicity:
c:=1: for i from 1 to n do c:=c*a(i)!*i^a(i): od:
prop:=nump^2;
s:=s + (n!/c)*prop;
od;
lprint(n, s);
A[n]:=s;
od:
[seq(A[n], n=1..M)];
# Alternatively after Reshetnikov:
a := n -> n!*(add(1/k, k=1..n)^2 + add(1/k-1/k^2, k=1..n)):
|
|
MATHEMATICA
|
Table[-Coefficient[CharacteristicPolynomial[ Array[KroneckerDelta[#1, #2]((((#1+1)))-1)+1&, {n, n}], x], x], {n, 1, 10}] (* John M. Campbell, May 24 2011 *)
Table[n! (HarmonicNumber[n] + HarmonicNumber[n]^2 - HarmonicNumber[n, 2]), {n, 1, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|