A006656 -id:A006656 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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.

Cf. A006656, A069853.

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

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

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 *)

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

[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

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

A093485 a(n) = (27n^2 + 9n + 2)/2.

Original entry on oeis.org

1, 19, 64, 136, 235, 361, 514, 694, 901, 1135, 1396, 1684, 1999, 2341, 2710, 3106, 3529, 3979, 4456, 4960, 5491, 6049, 6634, 7246, 7885, 8551, 9244, 9964, 10711, 11485, 12286, 13114, 13969, 14851, 15760, 16696, 17659, 18649, 19666, 20710, 21781

Offset: 0

Michael Joseph Halm, May 13 2004

Dodecahedral gnomon numbers: first differences of dodecahedral numbers.
The sequence is related to other gnomon numbers of polyhedra, known by other more familiar names: triangular numbers (tetrahedral gnomon numbers), hexagonal numbers (cubic gnomon numbers), square pyramidal numbers (octahedral gnomon numbers).
A124388 = first differences; second differences = 27. - Reinhard Zumkeller, Oct 30 2006
Sums of the triangular numbers from A000217(3*n-1) to A000217(3*n+1), with A000217(-1) = 0. - Bruno Berselli, Sep 04 2018

			a(1) = 19 because (1+1)*(3*(1+1)-1)*(3*(1+1)-2)/2-1*(3*1-1)*(3*1-2)/2 = 2*(6-1)*(6-2)/2 - 1*(3-1)*(3-2)/2 = 20-1 = 19.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000330, A003215, A005901, A006656.

a093485 n = (9 * n * (3 * n + 1) + 2) `div` 2
-- Reinhard Zumkeller, Jun 16 2013

[(27*n^2 + 9*n + 2)/2 : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011

a(n)=(27*n^2+9*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (n+1)*(3*(n+1)-1)*(3*(n+1)-2)/2-n*(3*n-1)*(3*n-2)/2.

G.f.: (1 + 16*x + 10*x^2)/(1 - x)^3. - Colin Barker, Mar 28 2012

New definition from Ralf Stephan, Dec 01 2004

Name corrected and initial term added by Arkadiusz Wesolowski, Aug 15 2011

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).

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

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A093485 a(n) = (27*n^2 + 9*n + 2)/2.

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

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Author

Keywords

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Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A093485 a(n) = (27n^2 + 9n + 2)/2.