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%I M5114 N2216 #55 Feb 03 2022 02:26:46
%S 1,21,322,4536,63273,902055,13339535,206070150,3336118786,56663366760,
%T 1009672107080,18861567058880,369012649234384,7551527592063024,
%U 161429736530118960,3599979517947607200,83637381699544802976,2021687376910682741568,50779532534302850198976,1323714091579185857760000
%N Unsigned Stirling numbers of first kind s(n,6).
%C The asymptotic expansion of the higher order exponential integral E(x,m=6,n=1) ~ exp(-x)/x^6*(1 - 21/x + 322/x^2 - 4536/x^3 + 63273/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).
%H T. D. Noe, <a href="/A001233/b001233.txt">Table of n, a(n) for n = 6..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 Let P(n-1,X) = (X+1)(X+2)(X+3)...(X+n-1); then a(n) is the coefficient of X^5; or a(n) = P'''''(n-1,0)/5!. - _Benoit Cloitre_, May 09 2002 [Edited by _Petros Hadjicostas_, Jun 29 2020 to agree with the offset of 6]
%F E.g.f.: (-log(1-x))^6/6!.
%F a(n) is coefficient of x^(n+6) in (-log(1-x))^6, multiplied by (n+6)!/6!.
%F a(n) = det(|S(i+6,j+5)|, 1 <= i,j <= n-6), where S(n,k) are Stirling numbers of the second kind. - _Mircea Merca_, Apr 06 2013
%e (-log(1-x))^6 = x^6 + 3*x^7 + 23/4*x^8 + 9*x^9 + ...
%t Drop[Abs[StirlingS1[Range[30],6]],5] (* _Harvey P. Dale_, Sep 17 2013 *)
%o (PARI) for(n=5,50,print1(polcoeff(prod(i=1,n,x+i),5,x),","))
%o (Sage) [stirling_number1(i,6) for i in range(6,22)] # _Zerinvary Lajos_, Jun 27 2008
%Y Cf. A000254, A000399, A000454, A000482, A001234, A008275, A243569, A243570.
%K nonn,easy
%O 6,2
%A _N. J. A. Sloane_
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