cp's OEIS Frontend

A007415 Expand sin x / exp x = x-x^2+x^3/3-x^5/30+... and invert nonzero coefficients.

%I A007415 #25 Jul 08 2025 00:59:24
%S A007415 0,1,-1,3,0,-30,90,-630,0,22680,-113400,1247400,0,-97297200,681080400,
%T A007415 -10216206000,0,1389404016000,-12504636144000,237588086736000,0,
%U A007415 -49893498214560000,548828480360160000,-12623055048283680000,0,3786916514485104000000
%N A007415 Expand sin x / exp x = x-x^2+x^3/3-x^5/30+... and invert nonzero coefficients.
%H A007415 J. Roberts, <a href="/A007415/a007415.pdf">Lure of the Integers</a>, Annotated scanned copy of pp. 81, 86 with notes.
%F A007415 a(n) = [n mod 4 > 0] * (-1)^(n+1+[n/4]) * n!/2^[n/2]. - _Ralf Stephan_, Mar 06 2004
%F A007415 E.g.f.: sin(x)/exp(x) = x-x^2/(G(0)+x); G(k)=2k+1-x+x*(2k+1)/(4k+3-x+x^2*(4k+3)/( (2k+2)*(4k+5)-x^2+x*(2k+2)*(4k+5)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 20 2011
%p A007415 a:= n-> (p-> `if`(p=0,0,1/p))(coeff(series(sin(x)/exp(x), x, n+1), x, n)):
%p A007415 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 17 2017
%t A007415 1/CoefficientList[Sin[x]/Exp[x] + O[x]^26, x] /. ComplexInfinity -> 0 // Quiet (* _Jean-François Alcover_, Feb 26 2019 *)
%Y A007415 Absolute values are essentially the same as A046979, where zeros are replaced by ones.
%Y A007415 a(4n+2) = -(-1)^n*A052277(n), a(2n+1) = (-1)^[n/2]*A007019(n).
%Y A007415 Cf. A046981, A007452, A009775, A092820.
%K A007415 sign
%O A007415 0,4
%A A007415 _N. J. A. Sloane_