0,3

Number of permutations of n elements with exactly two cycles.

a(n)=number of cycles in all permutations of [n]. Example: a(3)=11 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) have 11 cycles alltogether. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 12 2004

The sum of the top levels of the last column over all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=3 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, the levels of their last columns being 2 and 1, respectively. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 12 2006

a(n) is divisible by n for all composite n >= 6. a(2n) is divisible by (2n+1). - Leroy Quet May 20 2007

For n >= 2 the determinant of the n-1 X n-1 matrix M(i,j) = i + 2 for i = j and 1 otherwise (i,j = 1..n-1). E.g. for n = 3 the determinant of [(3, 1), (1, 4)]. See 53rd Putnam Examination, 1992, Problem B5. - Franz Vrabec (franz.vrabec(AT)aon.at), Jan 13 2008, Mar 26 2008

Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Jan 07 2009: (Start)

The numerator of the fraction when we sum (without simplification) the terms in the harmonic sequence.

(1+1/2=2/2+1/2= 3/2; 3/2+1/3=9/6+2/6= 11/6; 11/6+1/4=44/24+6/24= 50/24;...)

The denominator of this fraction is n! A000142.

(End)

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 2009: (Start)

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=1) ~ exp(-x)/x^2*(1 - 3/x + 11/x^2 - 50/x^3 + 274/x^4 - 1764/x^5 + 13068/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information.

(End)

Contribution from Tom Woodward (twoodward(AT)macalester.edu), Nov 12 2009: (Start)

a(n)=number of permutations of [n+1] containing exactly 2 cycles. Example: a(2)=3

because the permutations (1)(23), (12)(3), (13)(2) are the only permutations of [3]

with exactly 2 cycles. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identities 186-190.

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, 1986, see page 2. MR0863284 (89d:41049)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100

Leroy Quet, Home Page (listed in lieu of email address)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 31

J. Scholes, 53rd Putnam 1992, Problem B5.

Let P(n,X)=(X+1)(X+2)(X+3)...(X+n); then a(n) is the coefficient of X; or a(n)=P'(n,0) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2002

Sum_{n>0} a(n)x^n/n!^2 = exp(x)(Sum_{n>0}(-1)^(n+1)x^n/(n*n!)). - Michael Somos Mar 24 2004. Corrected by Warren Smith, Feb 12 2006.

a(n) is coefficient of x^(n+2) in (-log(1-x))^2, multiplied by (n+2)!/2.

Also a(n) = n!*Sum 1/i, i=1..n = n!*H(n), H(n) = harmonic number = A001008/A002805.

a(n) ~ 2^(1/2)*pi^(1/2)*log(n)*n^(1/2)*e^-n*n^n - Joe Keane (jgk(AT)jgk.org), Jun 06 2002

E.g.f.: log(1-x)/(x-1). (= (log(1-x))^2/2 if offset 1). - Michael Somos Feb 05 2004

a(n)=a(n-1)(2n-1)-a(n-2)(n-1)^2, if n>1. - Michael Somos Mar 24 2004

a(n)=A081358(n)+A092691(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 12 2004

a(n) = n!*Sqrt[Sum[Sum[1/(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 26 2004

a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 29 2005

p^2 divides a(p-1) for prime p>3. a(n) = Sum[ 1/i, {i,1,n}] / Product[ 1/i, {i,1,n}]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 11 2006

For n>=1, a(n)=n!*sum(1/(n-k),k=0..n-1); [From Milan R. Janjic (agnus(AT)blic.net), Dec 14 2008]

(1-x)^-1 * (-log(1-x)) = x + 3/2*x^2 + 11/6*x^3 + 25/12*x^4 + ...

A000254 := proc(n) option remember; if n<=1 then 1 else n*A000254(n-1)+(n-1)!; fi; end;

a:=n->sum(n!/k, k=1..n): seq(a(n), n=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 22 2008

Table[ (PolyGamma[ m ]+EulerGamma) (m-1)!, {m, 1, 24} ] - from W. Meeussen (wouter.meeussen(AT)pandora.be)

Table[ n!*HarmonicNumber[n], {n, 0, 19}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 21 2005)

Table[Sum[1/i, {i, 1, n}]/Product[1/i, {i, 1, n}], {n, 1, 30}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 11 2006

(MuPAD) A000254 := proc(n) begin n*A000254(n-1)+fact(n-1) end_proc: A000254(1) := 1:

(PARI) a(n)=if(n<0, 0, (n+1)!/2*sum(k=1, n, 1/k/(n+1-k)))

sage: [stirling_number1(i, 2) for i in xrange(1, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

Cf. A000399, A000774, A004041, A024167, A046674, A049034, A008275 (Stirling1 triangle).

Cf. A081358, A092691, A151881.

With signs: A081048.

Column 1 in triangle A008969.

Cf. A121633.

Sequence in context: A162477 A115081 A103466 this_sequence A081048 A065048 A024335

Adjacent sequences: A000251 A000252 A000253 this_sequence A000255 A000256 A000257

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).