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 A009551 #44 Feb 16 2025 08:32:32
%S A009551 0,1,2,5,20,101,606,4241,33928,305353,3053530,33588829,403065948,
%T A009551 5239857325,73358002550,1100370038249,17605920611984,299300650403729,
%U A009551 5387411707267122,102360822438075317,2047216448761506340
%N A009551 Expansion of sin(x)/(1-x).
%C A009551 a(n) equals the imaginary part of the permanent of the n X n matrix with (1+i)'s along the main diagonal, and 1's everywhere else. - _John M. Campbell_, Jul 10 2011
%H A009551 Vincenzo Librandi, <a href="/A009551/b009551.txt">Table of n, a(n) for n = 0..200</a>
%H A009551 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>.
%F A009551 a(n) = round(n!*sin(1)), n>=1. - _Vladeta Jovovic_, Aug 11 2002
%F A009551 a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k/(2k-1)!, n>0. - _Ralf Stephan_, Apr 16 2004
%F A009551 a(n) = n*a(n-1) - a(n-2) +(n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 07 2012
%F A009551 From _Vladimir Reshetnikov_, Oct 27 2015: (Start)
%F A009551 a(n) = Im(i^n*hypergeom([1,-n], [], i)).
%F A009551 a(n) = n!*sin(1)-cos(Pi*n/2)*hypergeom([1], [n/2+1,(n+3)/2], -1/4)/(n+1) + sin(Pi*n/2)*hypergeom([1], [n/2+2,(n+3)/2], -1/4)/(n^2+3*n+2).
%F A009551 a(n) = Im(Gamma(n+1, i)*exp(i)) = (Gamma(n+1, i)*exp(i)-Gamma(n+1, -i)*exp(-i))/(2*i), where Gamma(a, x) is the upper incomplete Gamma function, i=sqrt(-1).
%F A009551 Gamma(n+1, i) = exp(-i)*((-1)^n*A009102(n) + a(n)*i). (End)
%p A009551 restart: G(x):=sin(x)/(1-x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # _Zerinvary Lajos_, Apr 03 2009
%t A009551 Table[n!*SeriesCoefficient[Sin[x]/(1-x),{x,0,n}],{n,0,20}] (* corrected by _Vaclav Kotesovec_, Oct 07 2012 *)
%t A009551 With[{nn=30},CoefficientList[Series[Sin[x]/(1-x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Apr 17 2013 *)
%t A009551 Round@Table[Im[Gamma[n+1, I] E^I], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 27 2015 *)
%o A009551 (PARI) a(n) = round(n!*sin(1))
%o A009551 (Magma) I:=[1,2,5]; [0] cat [n le 3 select I[n] else n*Self(n-1)-Self(n-2)+(n-2)*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Jan 19 2018
%Y A009551 Cf. A009102, A000142, A000166, A000522, A000023, A053486, A010844 (incomplete Gamma function values at other points).
%K A009551 nonn
%O A009551 0,3
%A A009551 _R. H. Hardin_
%E A009551 More terms from _Benoit Cloitre_, Aug 13 2002