cp's OEIS Frontend

A000965 Numerators of expansion of e.g.f. sinh(x) / sin(x) (even powers only).

%I A000965 M1307 N0501 #38 Feb 01 2022 08:31:47
%S A000965 1,2,4,104,272,3104,79808,631936,1708288,7045156352,1413417032704,
%T A000965 6587672324096,37378439704576,66465881481076736,80812831866241024,
%U A000965 17004045797823707643904,55131841948562370265088,189924798793194975920128,1382061377731043599678963712
%N A000965 Numerators of expansion of e.g.f. sinh(x) / sin(x) (even powers only).
%D A000965 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000965 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000965 T. D. Noe, <a href="/A000965/b000965.txt">Table of n, a(n) for n = 0..100</a>
%H A000965 L. Carlitz, <a href="https://www.jstor.org/stable/3029628">The Coefficients of sinh x/sin x</a>, Mathematics Magazine, Vol. 29, No. 4 (Mar. - Apr., 1956), pp. 193-197.
%H A000965 J. M. Gandhi, <a href="http://www.jstor.org/stable/3029194">The coefficients of cosh x/cos x and a note on Carlitz's coefficients of sinh x/sin x</a>, Math. Mag., 31 (1958), 185-191.
%F A000965 Numerator of ( (2n)! times coefficient of x^(2n) in sinh x / sin x ). - corrected by _Sean A. Irvine_, Apr 17 2019
%e A000965 sinh(x)/sin(x) = 1 + 1/3*x^2 + 1/18*x^4 + 13/1890*x^6 + 17/22680*x^8 + ...
%p A000965 a:= n-> numer((2*n)!*coeff(series(sinh(x)/sin(x), x, 2*n+2), x, 2*n)):
%p A000965 seq(a(n), n=0..20);  # _Alois P. Heinz_, Feb 01 2022
%t A000965 nn = 42; t = Range[0, nn]! CoefficientList[Series[Sinh[x]/Sin[x], {x, 0, nn}], x]; t = Numerator[t]; Take[t, {1, nn, 2}] (* _T. D. Noe_, Jun 21 2012 *)
%o A000965 (PARI) my(x='x+O('x^40)); select(x->(x!=0),apply(x->numerator(x), Vec(serlaplace(sinh(x)/sin(x))))) \\ _Michel Marcus_, Apr 16 2019
%o A000965 (Magma)
%o A000965 m:=80; R<x>:=PowerSeriesRing(Rationals(), m);
%o A000965 b:= Coefficients(R!(Laplace( Sinh(x)/Sin(x) )));
%o A000965 [Numerator( b[2*n-1] ): n in [1..Floor((m-2)/2)]]; // _G. C. Greubel_, Jan 31 2022
%o A000965 (Sage) [numerator( factorial(2*n)*( sinh(x)/sin(x) ).series(x, 2*n+3).list()[2*n] ) for n in (0..60)] # _G. C. Greubel_, Jan 31 2022
%Y A000965 Cf. A006656, A069853.
%K A000965 nonn,easy
%O A000965 0,2
%A A000965 _N. J. A. Sloane_