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 A001233 M5114 N2216 #57 Feb 24 2025 09:13:33 %S A001233 1,21,322,4536,63273,902055,13339535,206070150,3336118786,56663366760, %T A001233 1009672107080,18861567058880,369012649234384,7551527592063024, %U A001233 161429736530118960,3599979517947607200,83637381699544802976,2021687376910682741568,50779532534302850198976,1323714091579185857760000 %N A001233 Unsigned Stirling numbers of first kind s(n,6). %C A001233 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 A001233 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 A001233 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226. %D A001233 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A001233 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A001233 T. D. Noe, <a href="/A001233/b001233.txt">Table of n, a(n) for n = 6..100</a> %H A001233 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 A001233 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 A001233 E.g.f.: (-log(1-x))^6/6!. %F A001233 a(n) is coefficient of x^(n+6) in (-log(1-x))^6, multiplied by (n+6)!/6!. %F A001233 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 %F A001233 a(n) = 3*(2*n - 7)*a(n-1) - 5*(3*n^2 - 24*n + 49)*a(n-2) + 10*(2*n - 9)*(n^2 - 9*n + 21)*a(n-3) - (15*n^4 - 300*n^3 + 2265*n^2 - 7650*n + 9751)*a(n-4) + (2*n - 11)*(n^2 - 11*n + 31)*(3*n^2 - 33*n + 91)*a(n-5) - (n-6)^6*a(n-6). - _Vaclav Kotesovec_, Feb 24 2025 %e A001233 (-log(1-x))^6 = x^6 + 3*x^7 + 23/4*x^8 + 9*x^9 + ... %t A001233 Drop[Abs[StirlingS1[Range[30],6]],5] (* _Harvey P. Dale_, Sep 17 2013 *) %o A001233 (PARI) for(n=5,50,print1(polcoeff(prod(i=1,n,x+i),5,x),",")) %o A001233 (Sage) [stirling_number1(i,6) for i in range(6,22)] # _Zerinvary Lajos_, Jun 27 2008 %Y A001233 Cf. A000254, A000399, A000454, A000482, A001234, A008275, A243569, A243570. %K A001233 nonn,easy %O A001233 6,2 %A A001233 _N. J. A. Sloane_