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 A053567 #51 Sep 08 2022 08:45:00
%S A053567 -120,1764,-13132,67284,-269325,902055,-2637558,6926634,-16669653,
%T A053567 37312275,-78558480,156952432,-299650806,549789282,-973941900,
%U A053567 1672280820,-2792167686,4546047198,-7234669596,11276842500,-17247104875,25922927745,-38343278610,55880640270
%N A053567 Stirling numbers of first kind, s(n+5, n).
%C A053567 a(n) is equal to (-1)^n times the sum of the products of each distinct grouping of 5 members of the set {1, 2, 3, ..., n + 4}. So, a(1) = (-1)*1*2*3*4*5 = -120, and a(2) = 1*2*3*4*5 + 1*2*3*4*6 + 1*2*3*5*6 + 1*2*4*5*6 + 1*3*4*5*6 + 2*3*4*5*6 = 1764. See comment at A001303. - _Greg Dresden_, Aug 26 2019
%D A053567 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.
%H A053567 Vincenzo Librandi, <a href="/A053567/b053567.txt">Table of n, a(n) for n = 1..200</a>
%H A053567 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 A053567 G. C. Greubel, <a href="https://arxiv.org/abs/1612.09385">A Note on Jain basis functions</a>, arXiv:1612.09385 [math.CA], 2016.
%H A053567 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (-11,-55,-165,-330,-462,-462,-330,-165,-55,-11,-1).
%F A053567 a(n) = (-1)^n*binomial(n+5, 6)*binomial(n+5, 2)*(3*n^2 + 23*n + 38)/8.
%F A053567 G.f.: -x*(120 - 444*x + 328*x^2 - 52*x^3 + x^4)/(1+x)^11. See row k=4 of triangle A112007 for the coefficients. [G.f. corrected by _Georg Fischer_, May 19 2019]
%F A053567 E.g.f. with offset 5: exp(x)*(Sum_{m=0..5} A112486(5, m)*(x^(5+m)/(5+m)!).
%F A053567 a(n) = (f(n+4, 5)/10!)*Sum_{m=0..min(5, n-1)} A112486(5, m)*f(10, 5-m)*f(n-1, m)), with the falling factorials f(n, m):=n*(n-1)*, ..., *(n-(m-1)). From the e.g.f.
%p A053567 A053567 := proc(n) (-1)^(n+1)*combinat[stirling1](n+5,n) ; end proc: # _R. J. Mathar_, Jun 08 2011
%t A053567 Table[StirlingS1[n+5,n](-1)^(n-1),{n,30}] (* _Harvey P. Dale_, Sep 21 2011 *)
%t A053567 (* or *)
%t A053567 CoefficientList[Series[-x*(120 - 444*x + 328*x^2 - 52*x^3 + x^4)/(1+x)^11, {x, 0, 27}], x] (* _Georg Fischer_, May 19 2019 *)
%o A053567 (Sage) [stirling_number1(n,n-5)*(-1)^(n+1) for n in range(6, 26)] # _Zerinvary Lajos_, May 16 2009
%o A053567 (Magma) [(-1)^n*Binomial(n+5, 6)*Binomial(n+5, 2)*(3*n^2+23*n+38)/8: n in [1..30]]; // _Vincenzo Librandi_, Jun 09 2011
%o A053567 (PARI) a(n) = (-1)^(n-1)*stirling(n+5, n, 1); \\ _Michel Marcus_, Aug 29 2017
%Y A053567 Next |Stirling1| diagonal A112002, 5th diagonal of A130534.
%K A053567 easy,sign
%O A053567 1,1
%A A053567 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
%E A053567 Definition edited by _Eric M. Schmidt_, Aug 29 2017
%E A053567 Incorrect formula removed by _Greg Dresden_, Aug 26 2019