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 A151881 Sum (number of cycles)^2 over all n! permutations of [1..n]. 6
 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 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) 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 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 *) CROSSREFS Cf. A000254, A151882. Sequence in context: A162815 A351756 A033312 * A229811 A359915 A362568 Adjacent sequences: A151878 A151879 A151880 * A151882 A151883 A151884 KEYWORD nonn AUTHOR N. J. A. Sloane, Jul 22 2009 STATUS approved

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Last modified October 1 14:33 EDT 2023. Contains 365826 sequences. (Running on oeis4.)