A003711 -id:A003711 - cp's OEIS frontend

A006229 Expansion of e.g.f. exp( tan x ).

1, 1, 1, 3, 9, 37, 177, 959, 6097, 41641, 325249, 2693691, 24807321, 241586893, 2558036145, 28607094455, 342232522657, 4315903789009, 57569080467073, 807258131578995, 11879658510739497, 183184249105857781, 2948163649552594737, 49548882107764546223

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259, Sum_{k} T(n,k).
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..480 (terms 0..100 from T. D. Noe)
J. Shallit, Letter to N. J. A. Sloane, May 1975
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Row sums of A059419 and unsigned A111593.

Cf. A003711, A003717, A000182, A047691, A047692.

function A006229_list(len::Int)
    len <= 0 && return BigInt[]
    T = zeros(BigInt, len, len); T[1,1] = 1
    S = Array(BigInt, len); S[1] = 1
    for n in 2:len
        T[n,n] = 1
        for k in 2:n-1 T[n,k] = T[n-1,k-1] + k*(k-1)*T[n-1,k+1] end
        S[n] = sum(T[n,k] for k in 2:n)
    end
S end
println(A006229_list(24)) # Peter Luschny, Apr 27 2017

With[{nn=30},CoefficientList[Series[Exp[Tan[x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 04 2011 *)

a(n):=sum(if oddp(n+k) then 0 else (-1)^((n+k)/2)*sum(j!/k!*stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1, k-1), j, k, n), k, 1, n); /* Vladimir Kruchinin, Aug 05 2010 */

E.g.f.: exp(tan(x)).
a(n) = sum(if oddp(n+k) then 0 else (-1)^((n+k)/2)*sum(j!/k!*stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n),k,1,n), n>0. - Vladimir Kruchinin, Aug 05 2010
E.g.f.: 1 + tan(x)/T(0), where T(k) = 4*k+1 - tan(x)/(2 + tan(x)/(4*k+3 - tan(x)/(2 + tan(x)/T(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
a(n) = Sum_{i=0..(n-1)/2} binomial(n-1,2*i)*z(i+1)*a(n-2*i-1), a(0)=1, where z(n) is tangent (or "zag") numbers (A000182). - Vladimir Kruchinin, Mar 04 2015 [corrected by Jason Yuen, Dec 29 2024]

More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001

A003710 Expansion of e.g.f. cos(tan(x)) (even powers only).

Original entry on oeis.org

1, -1, -7, -97, -2063, -53409, -752343, 166831871, 43685848289, 9398558916159, 2116926930779225, 524586454143030495, 144620290378876829905, 44287070229737735633567, 14954349885478653319004041

Offset: 0

R. H. Hardin, Simon Plouffe

sign

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

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A003709, A003711.

nn = 20; Table[(CoefficientList[Series[Cos[Tan[x]], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Feb 16 2015 *)

a(n):=if n=0 then 1 else 2*sum((sum(binomial(j-1,2*m-1)*j!*2^(2*n-j-1)*(-1)^(n+j)*stirling2(2*n,j),j,2*m,2*n))/(2*m)!,m,0,n); /* Vladimir Kruchinin, Jun 28 2011 */

my(x='x+O('x^30)); select(x->x, Vec(serlaplace(cos(tan(x))))) \\ Michel Marcus, Oct 02 2021

a(n) = 2 * Sum_{m=0..n} ( Sum_{j=2*m..2*n} binomial(j-1,2*m-1) * j! * 2^(2*n-j-1) * (-1)^(n+j) * Stirling2(2*n,j) )/(2*m)!, n>0, a(0)=1. - Vladimir Kruchinin, Jun 29 2011

A296856 Expansion of e.g.f. cosh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, 3, 15, 224, 4545, 126753, 4626076, 213703095, 12167727543, 835893746300, 68091766034061, 6483302813035857, 712860388963255000, 89585739948801890619, 12753524767335858733935, 2040804997678590563632568, 364567987004433619078313961

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			cosh(x*tan(x/2)) = 1 + 3*x^4/4! + 15*x^6/6! + 224*x^8/8! + 4545*x^10/10! + 126753*x^12/12! + ...

Cf. A001469, A003711, A009166, A110501, A296839, A296841, A296842, A296853, A296854.

nmax = 17; Table[(CoefficientList[Series[Cosh[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] cosh(x*tan(x/2)).

A013526 Numerator of [x^(2n)] of the Taylor expansion cosh(cosec(x)-cot(x))=1 +x^2/8 +3x^4/128 +59x^6/15360 +871*x^8/1474560 +....

Original entry on oeis.org

1, 1, 3, 59, 871, 325249, 35797, 24362249, 342232522657, 8224154352439, 23157229065769, 9926476934520521, 37638416003805990839, 296699416391356495667713, 691054566545631371393

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

The coefficients of the e.g.f. of x/2, cosh(cosec(x)-cot(x)) = 1 + 1*x^2/(2^2*2!) + 9*x^4/(2^4*4!) + 177*x^6/(2^6*6!) + 6097*x^8/(2^8*8!) + ..., are apparently the absolute values of A003711.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Numerator[Take[CoefficientList[Series[Cosh[Csc[x] - Cot[x]], {x, 0, 35}], x], {1, -2, 2}]] (* Vincenzo Librandi, Nov 12 2016 *)

Name edited by R. J. Mathar, Dec 19 2011

A298245 Expansion of e.g.f. exp(cos(tanh(x))-1) (even powers only).

Original entry on oeis.org

1, -1, 12, -327, 15883, -1202524, 130394253, -19113418989, 3632485387276, -867280709024131, 253803272212372575, -89250842789856565620, 37105568909251258810585, -17991614679286735149423193, 10057557723279565571532112044, -6417980557539322347015938082111

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

sign

			exp(cos(tanh(x))-1) = 1 - x^2/2! + 12*x^4/4! - 327*x^6/6! + 15883*x^8/8! - 1202524*x^10/10! + ...

Cf. A003711, A009090, A009201, A009202, A009203, A009204, A009238, A009239, A009240, A009241, A009254, A297214, A297215.

nmax = 15; Table[(CoefficientList[Series[Exp[Cos[Tanh[x]] - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] exp(cos(tanh(x))-1).

cp's OEIS Frontend

A006229 Expansion of e.g.f. exp( tan x ).

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A003710 Expansion of e.g.f. cos(tan(x)) (even powers only).

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A296856 Expansion of e.g.f. cosh(x*tan(x/2)) (even powers only).

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A013526 Numerator of [x^(2n)] of the Taylor expansion cosh(cosec(x)-cot(x))=1 +x^2/8 +3*x^4/128 +59*x^6/15360 +871*x^8/1474560 +....

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A298245 Expansion of e.g.f. exp(cos(tanh(x))-1) (even powers only).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A013526 Numerator of [x^(2n)] of the Taylor expansion cosh(cosec(x)-cot(x))=1 +x^2/8 +3x^4/128 +59x^6/15360 +871*x^8/1474560 +....