This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000482 M4983 N2142 #55 Feb 24 2025 09:13:03 %S A000482 1,15,175,1960,22449,269325,3416930,45995730,657206836,9957703756, %T A000482 159721605680,2706813345600,48366009233424,909299905844112, %U A000482 17950712280921504,371384787345228000,8037811822645051776,181664979520697076096,4280722865357147142912,105005310755917452984576 %N A000482 Unsigned Stirling numbers of first kind s(n,5). %C A000482 Number of permutations of n elements with exactly 5 cycles. %C A000482 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 A000482 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 A000482 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 A000482 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226. %D A000482 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000482 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000482 Shanzhen Gao, Permutations with Restricted Structure (in preparation) [_Shanzhen Gao_, Sep 14 2010] %H A000482 T. D. Noe, <a href="/A000482/b000482.txt">Table of n, a(n) for n=5..100</a> %H A000482 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 A000482 E.g.f.: (-log(1-x))^5/5!. [Corrected by _Joerg Arndt_, Oct 05 2009] %F A000482 a(n) is coefficient of x^(n+5) in (-log(1-x))^5, multiplied by (n+5)!/5!. %F A000482 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] %F A000482 a(n) = 5*(n-3)*a(n-1) - 5*(2*n^2 - 14*n + 25)*a(n-2) + 5*(n-4)*(2*n^2 - 16*n + 33)*a(n-3) - (5*n^4 - 90*n^3 + 610*n^2 - 1845*n + 2101)*a(n-4) + (n-5)^5*a(n-5). - _Vaclav Kotesovec_, Feb 24 2025 %e A000482 (-log(1-x))^5 = x^5 + 5/2*x^6 + 25/6*x^7 + 35/6*x^8 + ... %t A000482 Abs[StirlingS1[Range[5,30],5]] (* _Harvey P. Dale_, May 26 2014 *) %o A000482 (PARI) for(n=4,50,print1(polcoeff(prod(i=1,n,x+i),4,x),",")) %o A000482 (Sage) [stirling_number1(i,5) for i in range(5,22)] # _Zerinvary Lajos_, Jun 27 2008 %Y A000482 Cf. A000254, A000399, A000454, A001233, A001234, A008275, A243569, A243570. %K A000482 nonn,easy %O A000482 5,2 %A A000482 _N. J. A. Sloane_