A151881 Sum (number of cycles)^2 over all n! permutations of [1..n].

1, 5, 23, 120, 724, 5012, 39332, 345832, 3371976, 36135792, 422379792, 5349561984, 72996193152, 1067779243008, 16670798231040, 276718772067840, 4866610479828480, 90401487246167040, 1768784607499944960, 36360467544043008000, 783508616506603008000

Offset: 1

Author

N. J. A. Sloane, Jul 22 2009

Keywords

• nonn

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

• N. J. A. Sloane and Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 30 terms from N. J. A. Sloane)

Crossrefs

Cf. A000254, A151882.

Programs

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)):
seq(a(n), n=1..19); # Peter Luschny, Oct 21 2015

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 *)

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

E.g.f.: (log(1-x)^2-log(1-x))/(1-x). - Vladimir Reshetnikov, Oct 20 2015

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*k^2. - Peter Luschny, Jan 14 2020