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%I M4218 N1762 #97 Dec 18 2023 09:53:56
%S 1,6,35,225,1624,13132,118124,1172700,12753576,150917976,1931559552,
%T 26596717056,392156797824,6165817614720,102992244837120,
%U 1821602444624640,34012249593822720,668609730341153280,13803759753640704000
%N Unsigned Stirling numbers of first kind s(n,3).
%C Number of permutations of n elements with exactly 3 cycles.
%C 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. - _Johannes W. Meijer_, Oct 20 2009
%D 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 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%D Shanzhen Gao, Permutations with Restricted Structure (in preparation). - _Shanzhen Gao_, Sep 14 2010
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe and Robert Israel, <a href="/A000399/b000399.txt">Table of n, a(n) for n = 3..412</a> (3..100 from T. D. Noe)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=32">Encyclopedia of Combinatorial Structures 32</a>.
%H Sergey Kitaev and Jeffrey Remmel, <a href="http://arxiv.org/abs/1201.1323">Simple marked mesh patterns</a>, arXiv:1201.1323 [math.CO], 2012.
%H Sergey Kitaev and Jeffrey Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kitaev/kitaev5.html">Quadrant Marked Mesh Patterns</a>, J. Int. Seq. 15 (2012), #12.4.7.
%F 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_, May 09 2002 [Edited by _Petros Hadjicostas_, Jun 29 2020 to agree with the offset 3]
%F E.g.f.: -log(1-x)^3/3!.
%F a(n) is the coefficient of x^(n+3) in (-log(1-x))^3, multiplied by (n+3)!/6.
%F a(n) = ((Sum_{i=1..n-1} 1/i)^2 - Sum_{i=1..n-1} 1/i^2)*(n-1)!/2 for n >= 3. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000
%F a(n) = det(|S(i+3,j+2)|, 1 <= i,j <= n-3), where S(n,k) are Stirling numbers of the second kind. - _Mircea Merca_, Apr 06 2013
%F a(n) = Gamma(n)*(HarmonicNumber(n-1)^2 + Zeta(2,n) - Zeta(2))/2. - _Gerry Martens_, Jul 05 2015
%F From _Petros Hadjicostas_, Jun 28 2020: (Start)
%F a(n) = (n-3)! + (2*n-3)*a(n-1) - (n-2)^2*a(n-2) for n >= 5.
%F a(n) = 3*(n-2)*a(n-1) - (3*n^2-15*n+19)*a(n-2) + (n-3)^3*a(n-3) for n >= 6. (End)
%e (-log(1-x))^3 = x^3 + 3/2*x^4 + 7/4*x^5 + 15/8*x^6 + ...
%p seq(abs(Stirling1(n,3)),n=3..30); # _Robert Israel_, Jul 05 2015
%t a=Log[1/(1-x)];Range[0,20]! CoefficientList[Series[a^3/3!,{x,0,20}],x]
%t f[n_] := Abs@ StirlingS1[n, 3]; Array[f, 19, 3]
%t Abs[StirlingS1[Range[3,30],3]] (* _Harvey P. Dale_, Jun 23 2014 *)
%t f[n_] := Gamma[n]*(HarmonicNumber[n - 1]^2 + Zeta[2, n] - Zeta[2])/2; Array[f, 19, 3] (* _Robert G. Wilson v_, Jul 05 2015 *)
%o (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 (PARI) for(n=2,50,print1(polcoeff(prod(i=1,n,x+i),2,x),","))
%o (Sage) [stirling_number1(i+2,3) for i in range(1,22)] # _Zerinvary Lajos_, Jun 27 2008
%o (Magma) A000399:=func< n | Abs(StirlingFirst(n, 3)) >; [ A000399(n): n in [3..25] ]; // _Klaus Brockhaus_, Jan 14 2011
%Y Cf. A000254, A000454, A000482, A001233, A001234, A008275, A243569, A243570.
%K nonn,easy,nice
%O 3,2
%A _N. J. A. Sloane_
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