A047691 -id:A047691 - 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

A047692 Denominators of coefficients in Taylor series for exp(tan(x)).

Original entry on oeis.org

1, 1, 2, 2, 8, 120, 240, 720, 5760, 362880, 3628800, 57600, 691200, 6227020800, 830269440, 261534873600, 20922789888000, 355687428096000, 914624815104000, 426824913715200, 4742499041280000, 51090942171709440000, 3784514234941440000

Offset: 0

N. J. A. Sloane

			1 + 1*x + (1/2)*x^2 + (1/2)*x^3 + (3/8)*x^4 + (37/120)*x^5 + (59/240)*x^6 + (137/720)*x^7 + (871/5760)*x^8 + ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.

Cf. A047691, A006229.

A013516 Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3x^4/128 + 37x^5/3840 + 59*x^6/15360 + ...

Original entry on oeis.org

1, 2, 8, 16, 128, 3840, 15360, 92160, 1474560, 185794560, 3715891200, 117964800, 2831155200, 51011754393600, 13603134504960, 8569974738124800, 1371195958099968000, 46620662575398912000, 239763407530622976000

Offset: 0

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

The numerators are apparently the same as A047691.

			exp(cosec(x)-cot(x)) = 1 +1*x/(2^1*1!) + 1*x^2/(2^2*2!) + 3*x^3/(2^3*3!) + 9*x^4/(2^4*4!) + 37*x^5/(2^5*5!) +  177*x^6/(2^6*6!) +959*x^7/(2^7*7!)+ ...

Cf. A006229, A002425 (expansion of cosec(x)-cot(x)).

A013516 := proc(n)
        exp(csc(x)-cot(x)) ;
        coeftayl( %,x=0,n) ;
        denom(%) ;
end proc:  # R. J. Mathar, Dec 18 2011

a(n) = A047692(n) * 2^n. - Sean A. Irvine, Aug 07 2018

Corrected by R. J. Mathar, Dec 18 2011

A013522 Numerator of [x^(2n+1)] in the Taylor expansion sinh(cosec(x)-cotan(x))= x/2 +x^3/16 +37x^5/3840 +137x^7/92160 +41641x^9/185794560 + 3887x^11/117964800 +...

Original entry on oeis.org

1, 1, 37, 137, 41641, 3887, 241586893, 5721418891, 4315903789009, 2832484672207, 183184249105857781, 2154299222076719401, 1431144441595717024523, 386845480523042818420133, 21349170171172632123182767, 38112676874301043070814698873, 25659732417088795005806537367241

Offset: 0

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

Apparently a bisection of A047691.

The numerators in the e.g.f. of x/2, sinh(cosec(x)-cotan(x)) = x/(2^1*1!) +3*x^3/(2^3*3!) +37*x^5/(2^5*5!) +959*x^7/(2^7*7!) +41641*x^9/(2^9*9!)+.. are apparently covered by the absolute values of A003717.

G. C. Greubel, Table of n, a(n) for n = 0..200

Numerator[Take[CoefficientList[Series[Sinh[Csc[x] - Cot[x]], {x,0,45}], x], {2, -1, 2}]] (* G. C. Greubel, Nov 12 2016 *)

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

cp's OEIS Frontend

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Julia

Mathematica

Maxima

Formula

Extensions

A047692 Denominators of coefficients in Taylor series for exp(tan(x)).

Views

Author

Keywords

Examples

References

Crossrefs

A013516 Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3*x^4/128 + 37*x^5/3840 + 59*x^6/15360 + ...

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A013522 Numerator of [x^(2n+1)] in the Taylor expansion sinh(cosec(x)-cotan(x))= x/2 +x^3/16 +37*x^5/3840 +137*x^7/92160 +41641*x^9/185794560 + 3887*x^11/117964800 +...

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Extensions

A013516 Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3x^4/128 + 37x^5/3840 + 59*x^6/15360 + ...

A013522 Numerator of [x^(2n+1)] in the Taylor expansion sinh(cosec(x)-cotan(x))= x/2 +x^3/16 +37x^5/3840 +137x^7/92160 +41641x^9/185794560 + 3887x^11/117964800 +...