A000965 Numerators of expansion of e.g.f. sinh(x) / sin(x) (even powers only). (Formerly M1307 N0501)

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

Offset

0,2

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.

Formula

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

Example

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

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 *)

Prog

(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
(Magma)
m:=80; R<x>:=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
(SageMath) [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

Crossrefs

Cf. A006656, A069853.

Sequence in context: A166094 A018440 A104050 * A266176 A018449 A156506

Adjacent sequences: A000962 A000963 A000964 * A000966 A000967 A000968

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved