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A000965 Numerators of expansion of e.g.f. sinh(x) / sin(x) (even powers only).
(Formerly M1307 N0501)
3
1, 2, 4, 104, 272, 3104, 79808, 631936, 1708288, 7045156352, 1413417032704, 6587672324096, 37378439704576, 66465881481076736, 80812831866241024, 17004045797823707643904, 55131841948562370265088, 189924798793194975920128, 1382061377731043599678963712 (list; graph; refs; listen; history; text; internal format)
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
L. Carlitz, The Coefficients of sinh x/sin x, Mathematics Magazine, Vol. 29, No. 4 (Mar. - Apr., 1956), pp. 193-197.
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
(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
CROSSREFS
Sequence in context: A166094 A018440 A104050 * A266176 A018449 A156506
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)