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%I M4730 N2022 #85 Dec 19 2021 10:57:40
%S 1,10,85,735,6769,67284,723680,8409500,105258076,1414014888,
%T 20313753096,310989260400,5056995703824,87077748875904,
%U 1583313975727488,30321254007719424,610116075740491776
%N Unsigned Stirling numbers of first kind s(n,4).
%C Number of permutations of n elements with exactly 4 cycles.
%C The asymptotic expansion of the higher order exponential integral E(x, m=4, n=1) ~ exp(-x)/x^4*(1 - 10/x + 85/x^2 - 735/x^3 + 6769/x^4 - ...) leads to the sequence given above. See A163931 and A163932 for more information. - _Johannes W. Meijer_, Jun 11 2016
%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 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).
%D Shanzhen Gao, Permutations with Restricted Structure (in preparation) [From _Shanzhen Gao_, Sep 14 2010] [Apparently unpublished as of June 2016]
%H T. D. Noe, <a href="/A000454/b000454.txt">Table of n, a(n) for n=4..100</a>
%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=33">Encyclopedia of Combinatorial Structures 33</a>.
%F Let P(n-1,X) = (X+1)(X+2)(X+3)...(X+n-1); then a(n) is the coefficient of X^3; or a(n) = P'''(n-1,0)/3!. - _Benoit Cloitre_, May 09 2002 [Edited by _Petros Hadjicostas_, Jun 29 2020 to agree with the offset 4]
%F E.g.f.: (-log(1-x))^4/4!. [Corrected by _Joerg Arndt_, Oct 05 2009]
%F a(n) is coefficient of x^(n+4) in (-log(1-x))^4, multiplied by (n+4)!/4!.
%F a(n) = (h(n-1, 1)^3 - 3*h(n-1, 1)*h(n-1, 2) + 2*h(n-1, 3))*(n-1)!/3!, where h(n, r) = Sum_{i=1..n} 1/i^r. - Klaus Strassburger, 2000
%F a(n) = det(|S(i+4,j+3)|, 1 <= i,j <= n-4), where S(n,k) are Stirling numbers of the second kind. - _Mircea Merca_, Apr 06 2013
%F a(n) = y(n)*n!/24, where y(0) = y(1) = y(2) = y(3) = 0, y(4) = 1 and n^4*y(n) + (-1-5*n-10*n^2-10*n^3-4*n^4)*y(n+1) + (1+n)*(2+n)*(7+12*n+6*n^2)*y(n+2) - 2*(1+n)*(2+n)*(3+n)*(3+2*n)*y(3+n) + (1+n)*(2+n)*(3+n)*(4+n)*y(n+4) = 0. - _Benedict W. J. Irwin_, Jul 12 2016
%F From _Vaclav Kotesovec_, Jul 12 2016: (Start)
%F a(n) = 2*(2*n - 5)*a(n-1) - (6*n^2 - 36*n + 55)*a(n-2) + (2*n - 7)*(2*n^2 - 14*n + 25)*a(n-3) - (n-4)^4*a(n-4).
%F a(n) ~ n! * (log(n))^3 / (6*n) * (1 + 3*gamma/log(n) + (3*gamma^2 - Pi^2/2)/ (log(n))^2), where gamma is the Euler-Mascheroni constant A001620. (End)
%F From _Petros Hadjicostas_, Jun 29 2020: (Start)
%F a(n) = A000399(n-1) + (n-1)*a(n-1) for n >= 1 (assuming a(n) = 0 for n = 0..3).
%F a(n) = A103719(n-4) + (n-2)*a(n-1) for n >= 4.
%F a(n) = A000254(n-3) + (2*n-3)*a(n-1) - (n-2)^2*a(n-2) for n >= 3.
%F a(n) = (n-4)! + 3*(n-2)*a(n-1) - (3*n^2-15*n+19)*a(n-2) + (n-3)^3*a(n-3) for n >= 4. (End)
%e (-log(1-x))^4 = x^4 + 2*x^5 + (17/6)*x^6 + (7/2)*x^7 + ...
%t Abs[StirlingS1[Range[4,20],4]] (* _Harvey P. Dale_, Aug 26 2011 *)
%o (PARI) for(n=3,50,print1(polcoeff(prod(i=1,n,x+i),3,x),","))
%o (Sage) [stirling_number1(i,4) for i in range(4,22)] # _Zerinvary Lajos_, Jun 27 2008
%Y Cf. A000254, A000399, A000482, A001233, A001234, A008275, A243569, A243570.
%K nonn
%O 4,2
%A _N. J. A. Sloane_
%E More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000
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