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%I A000254 M2902 N1165
%S A000254 0,1,3,11,50,274,1764,13068,109584,1026576,10628640,120543840,
%T A000254 1486442880,19802759040,283465647360,4339163001600,70734282393600,
%U A000254 1223405590579200,22376988058521600,431565146817638400
%N A000254 Unsigned Stirling numbers of first kind, s(n+1,2): a(n+1)=(n+1)*a(n)+n!.
%C A000254 Number of permutations of n elements with exactly two cycles.
%C A000254 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
%C A000254 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
%C A000254 a(n) is divisible by n for all composite n >= 6. a(2n) is divisible by 
               (2n+1). - Leroy Quet May 20 2007
%C A000254 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)planetuniqa.at), Jan 13 2008, Mar 
               26 2008
%C A000254 Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Jan 07 2009: 
               (Start)
%C A000254 The numerator of the fraction when we sum (without simplification) the 
               terms in the harmonic sequence.
%C A000254 (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;
               ...)
%C A000254 The denominator of this fraction is n! A000142.
%C A000254 (End)
%C A000254 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 
               2009: (Start)
%C A000254 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.
%C A000254 (End)
%C A000254 Contribution from Tom Woodward (twoodward(AT)macalester.edu), Nov 12 
               2009: (Start)
%C A000254 a(n)=number of permutations of [n+1] containing exactly 2 cycles. Example: 
               a(2)=3
%C A000254 because the permutations (1)(23), (12)(3), (13)(2) are the only permutations 
               of [3]
%C A000254 with exactly 2 cycles. (End)
%D A000254 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.
%D A000254 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations 
               and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%D A000254 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, identities 186-190.
%D A000254 N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 
               Dover Publications, 1986, see page 2. MR0863284 (89d:41049)
%D A000254 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
%D A000254 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 226.
%D A000254 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000254 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000254 T. D. Noe, <a href="b000254.txt">Table of n, a(n) for n=0..100</a>
%H A000254 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> 
               (listed in lieu of email address)
%H A000254 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A000254 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=31">
               Encyclopedia of Combinatorial Structures 31</a>
%H A000254 J. Scholes, <a href="http://www.kalva.demon.co.uk/putnam/psoln/psol9211.html">
               53rd Putnam 1992, Problem B5</a>.
%F A000254 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
%F A000254 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.
%F A000254 a(n) is coefficient of x^(n+2) in (-log(1-x))^2, multiplied by (n+2)!/
               2.
%F A000254 Also a(n) = n!*Sum 1/i, i=1..n = n!*H(n), H(n) = harmonic number = A001008/
               A002805.
%F A000254 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
%F A000254 E.g.f.: log(1-x)/(x-1). (= (log(1-x))^2/2 if offset 1). - Michael Somos 
               Feb 05 2004
%F A000254 a(n)=a(n-1)(2n-1)-a(n-2)(n-1)^2, if n>1. - Michael Somos Mar 24 2004
%F A000254 a(n)=A081358(n)+A092691(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 12 2004
%F A000254 a(n) = n!*Sqrt[Sum[Sum[1/(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk 
               (alex(AT)kolmogorov.com), Oct 26 2004
%F A000254 a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Jan 29 2005
%F A000254 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
%F A000254 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]
%e A000254 (1-x)^-1 * (-log(1-x)) = x + 3/2*x^2 + 11/6*x^3 + 25/12*x^4 + ...
%p A000254 A000254 := proc(n) option remember; if n<=1 then 1 else n*A000254(n-1)+(n-1)!; 
               fi; end;
%p A000254 a:=n->sum(n!/k, k=1..n): seq(a(n), n=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 22 2008
%t A000254 Table[ (PolyGamma[ m ]+EulerGamma) (m-1)!, {m, 1, 24} ] - from W. Meeussen 
               (wouter.meeussen(AT)pandora.be)
%t A000254 Table[ n!*HarmonicNumber[n], {n, 0, 19}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), 
               May 21 2005)
%t A000254 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
%o A000254 (MuPAD) A000254 := proc(n) begin n*A000254(n-1)+fact(n-1) end_proc: A000254(1) 
               := 1:
%o A000254 (PARI) a(n)=if(n<0,0,(n+1)!/2*sum(k=1,n,1/k/(n+1-k)))
%o A000254 sage: [stirling_number1(i,2) for i in xrange(1,22)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 27 2008
%Y A000254 Cf. A000399, A000774, A004041, A024167, A046674, A049034, A008275 (Stirling1 
               triangle).
%Y A000254 Cf. A081358, A092691, A151881.
%Y A000254 With signs: A081048.
%Y A000254 Column 1 in triangle A008969.
%Y A000254 Cf. A121633.
%Y A000254 Sequence in context: A162477 A115081 A103466 this_sequence A081048 A065048 
               A024335
%Y A000254 Adjacent sequences: A000251 A000252 A000253 this_sequence A000255 A000256 
               A000257
%K A000254 nonn,easy,nice,new
%O A000254 0,3
%A A000254 N. J. A. Sloane (njas(AT)research.att.com).

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