Search: id:A000399
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%I A000399 M4218 N1762
%S A000399 1,6,35,225,1624,13132,118124,1172700,12753576,150917976,1931559552,
%T A000399 26596717056,392156797824,6165817614720,102992244837120,1821602444624640,
%U A000399 34012249593822720,668609730341153280,13803759753640704000
%N A000399 Unsigned Stirling numbers of first kind s(n,3).
%C A000399 Number of permutations of n elements with exactly 3 cycles.
%C A000399 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 
               2009: (Start)
%C A000399 The asymptotic expansion of the higher order exponential integral E(x,
               m=3,n=1) ~ exp(-x)/x^3*(1 - 6/x + 35/x^2 - 225/x^3 + 1624/x^4 - 13132/
               x^5 + ...) leads to the sequence given above. See A163931 and A163932 
               for more information.
%C A000399 (End)
%D A000399 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 A000399 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
%D A000399 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 226.
%D A000399 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000399 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000399 T. D. Noe, <a href="b000399.txt">Table of n, a(n) for n=3..100</a>
%H A000399 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 A000399 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=32">
               Encyclopedia of Combinatorial Structures 32</a>
%F A000399 Let P(n+1,X)=(X+1)(X+2)(X+3)...(X+n+1); then a(n) is the coefficient 
               of X^2; or a(n)=P''(n+1,0)/2! - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 09 2002
%F A000399 E.g.f. (-log(1-x))^3/3! or (1-x)^(-1) * (-log(1-x))^2. [Corrected by 
               Joerg Arndt, Oct 05 2009]
%F A000399 a(n) is coefficient of x^(n+3) in (-log(1-x))^3, multiplied by (n+3)!/
               6.
%F A000399 [(sum(1/i, i=1..n-1)^2-sum(1/i^2, i=1..n-1)]*(n-1)!/2 -Klaus Strassburger 
               (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000
%e A000399 (-log(1-x))^3 = x^3 + 3/2*x^4 + 7/4*x^5 + 15/8*x^6 + ...
%o A000399 (MuPAD) f := proc(n) option remember; begin n^3*f(n-3)-(3*n^2+3*n+1)*f(n-2)+3*(n+1)*f(n-1) 
               end_proc: f(0) := 1: f(1) := 6: f(2) := 35:
%o A000399 (PARI) for(n=2,50,print1(polcoeff(prod(i=1,n,x+i),2,x),","))
%o A000399 sage: [stirling_number1(i+2,3) for i in xrange(1,22)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 27 2008
%Y A000399 Cf. A000254, A000454, A000482, A001233, A008275 (Stirling1 triangle).
%Y A000399 Sequence in context: A001109 A144638 A117671 this_sequence A081051 A145145 
               A087631
%Y A000399 Adjacent sequences: A000396 A000397 A000398 this_sequence A000400 A000401 
               A000402
%K A000399 nonn,easy,nice
%O A000399 3,2
%A A000399 N. J. A. Sloane (njas(AT)research.att.com).

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