A000965 -id:A000965 - cp's OEIS frontend

A006656 Denominators of expansion of sinh x / sin x.

1, 3, 3, 21, 9, 11, 21, 9, 1, 133, 693, 69, 7, 189, 3, 7161, 231, 7, 399, 63, 77, 3311, 4347, 987, 49, 33, 33, 627, 57, 59, 7161, 2079, 11, 10787, 207, 2343, 1463, 4389, 231, 1659, 6237, 913, 9933, 693, 161, 7301833, 19184319, 4389, 11, 99, 33

Offset: 0

N. J. A. Sloane, Simon Plouffe

Also denominators of the expansion of tan(x)/tanh(x). - G. C. Greubel, Jan 31 2022

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
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.

Cf. A000965, A296628.

m:=120; R:=PowerSeriesRing(Rationals(), m);
b:= Coefficients(R!(Laplace( Sinh(x)/Sin(x) )));
[Denominator( b[2*n-1] ): n in [1..Floor((m-2)/2)]]; // G. C. Greubel, Jan 31 2022

With[{nn=100},Denominator[Take[CoefficientList[Series[Sinh[x]/Sin[x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]]] (* Harvey P. Dale, Jul 10 2013 *)

[denominator( 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

More terms from Harvey P. Dale, Jul 10 2013

A296628 Numerators of coefficients in expansion of e.g.f. tan(x)/tanh(x) (even powers only).

Original entry on oeis.org

1, 4, 16, 1408, 13568, 606208, 61878272, 1956380672, 21143027712, 348742016303104, 279852224852525056, 5217315235815227392, 118411884225053589504, 842233813811702133686272, 4096134057254358725165056, 3447514330976633343761929207808, 44711197753944482628093599547392

Offset: 0

Ilya Gutkovskiy, Dec 17 2017

The values of the denominators are similar to A006656 up to n = 138, but differ after.

			tan(x)/tanh(x) = 1 + (4/3)*x^2/2! + (16/3)*x^4/4! + (1408/21)*x^6/6! + (13568/9)*x^8/8! + ...

G. C. Greubel, Table of n, a(n) for n = 0..150

Cf. A000182, A000795, A000965, A006656, A024342.

m:=50; R:=PowerSeriesRing(Rationals(), m);
b:= Coefficients(R!(Laplace( Tan(x)/Tanh(x) )));
[Numerator( b[2*n-1] ): n in [1..Floor((m-2)/2)]]; // G. C. Greubel, Jan 31 2022

nmax = 16; Numerator[Table[(CoefficientList[Series[Tan[x]/Tanh[x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]]

[numerator( factorial(2*n)*( tan(x)/tanh(x) ).series(x, 2*n+3).list()[2*n] ) for n in (0..40)] # G. C. Greubel, Jan 31 2022

Numerators of coefficients in expansion of e.g.f. tanh(x)/tan(x) (even powers only, absolute values).

Numerators of coefficients in expansion of e.g.f. sin(x)*cosh(x)/(sinh(x)*cos(x)) (even powers only).

cp's OEIS Frontend

A006656 Denominators of expansion of sinh x / sin x.

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