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

1, 2, 4, 104, 272, 3104, 79808, 631936, 1708288, 7045156352, 1413417032704, 6587672324096, 37378439704576, 66465881481076736, 80812831866241024, 17004045797823707643904, 55131841948562370265088, 189924798793194975920128, 1382061377731043599678963712

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			sinh(x)/sin(x) = 1 + 1/3*x^2 + 1/18*x^4 + 13/1890*x^6 + 17/22680*x^8 + ...
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• L. Carlitz, The Coefficients of sinh x/sin x, Mathematics Magazine, Vol. 29, No. 4 (Mar. - Apr., 1956), pp. 193-197.
• J. M. Gandhi, The coefficients of cosh x/cos x and a note on Carlitz's coefficients of sinh x/sin x, Math. Mag., 31 (1958), 185-191.

Crossrefs

Cf. A006656, A069853.

Programs

Magma

m:=80; R:=PowerSeriesRing(Rationals(), m);
b:= Coefficients(R!(Laplace( Sinh(x)/Sin(x) )));
[Numerator( b[2*n-1] ): n in [1..Floor((m-2)/2)]]; // G. C. Greubel, Jan 31 2022

Maple

a:= n-> numer((2*n)!*coeff(series(sinh(x)/sin(x), x, 2*n+2), x, 2*n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 01 2022

Mathematica

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 *)

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

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

Formula

Numerator of ( (2n)! times coefficient of x^(2n) in sinh x / sin x ). - corrected by Sean A. Irvine, Apr 17 2019