%I M4983 N2142 #53 Aug 11 2024 23:23:47
%S 1,15,175,1960,22449,269325,3416930,45995730,657206836,9957703756,
%T 159721605680,2706813345600,48366009233424,909299905844112,
%U 17950712280921504,371384787345228000,8037811822645051776,181664979520697076096,4280722865357147142912,105005310755917452984576
%N Unsigned Stirling numbers of first kind s(n,5).
%C Number of permutations of n elements with exactly 5 cycles.
%C Let P(n-1,X) = (X+1)(X+2)(X+3)...(X+n-1); then a(n) is the coefficient of X^4; or a(n) = P''''(n-1,0)/4! - _Benoit Cloitre_, May 09 2002 [Edited by _Petros Hadjicostas_, Jun 29 2020 to agree with the offset of 5]
%C The asymptotic expansion of the higher order exponential integral E(x,m=5,n=1) ~ exp(-x)/x^5*(1 - 15/x + 175/x^2 - 1960/x^3 + 22449/x^4 - ...) leads to the sequence given above. See A163931 for E(x,m,n) information and A163932 for a Maple procedure for the asymptotic expansion. - _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 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) [_Shanzhen Gao_, Sep 14 2010]
%H T. D. Noe, <a href="/A000482/b000482.txt">Table of n, a(n) for n=5..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].
%F E.g.f.: (-log(1-x))^5/5!. [Corrected by _Joerg Arndt_, Oct 05 2009]
%F a(n) is coefficient of x^(n+5) in (-log(1-x))^5, multiplied by (n+5)!/5!.
%F a(n) = det(|S(i+5,j+4)|, 1 <= i,j <= n-5), where S(n,k) are Stirling numbers of the second kind. [_Mircea Merca_, Apr 06 2013]
%e (-log(1-x))^5 = x^5 + 5/2*x^6 + 25/6*x^7 + 35/6*x^8 + ...
%t Abs[StirlingS1[Range[5,30],5]] (* _Harvey P. Dale_, May 26 2014 *)
%o (PARI) for(n=4,50,print1(polcoeff(prod(i=1,n,x+i),4,x),","))
%o (Sage) [stirling_number1(i,5) for i in range(5,22)] # _Zerinvary Lajos_, Jun 27 2008
%Y Cf. A000254, A000399, A000454, A001233, A001234, A008275, A243569, A243570.
%K nonn,easy
%O 5,2
%A _N. J. A. Sloane_