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A151881
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Sum (number of cycles)^2 over all n! permutations of [1..n].
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5
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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; internal format)
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OFFSET
| 1,2
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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]
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 1..30
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FORMULA
| a(n) = (-1)^(n+1)*(Stirling1(n+1,2)-2*Stirling1(n+1,3)). [From Vladeta Jovovic (vladeta(AT)eunet.yu), Jul 22 2009]
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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)];
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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 *)
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CROSSREFS
| Cf. A000254, A151882.
Sequence in context: A193704 A162815 A033312 * A121636 A200028 A020032
Adjacent sequences: A151878 A151879 A151880 * A151882 A151883 A151884
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jul 22 2009
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