cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A318000 E.g.f.: log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).

Original entry on oeis.org

1, 4, 24, 256, 3840, 73024, 1688064, 45991936, 1443102720, 51249316864, 2032187080704, 89000317321216, 4266655914393600, 222232483747938304, 12496860570760249344, 754582425618372100096, 48694058763984285204480, 3344368871374116303929344, 243577066332044464943529984, 18751361596512920229250072576
Offset: 1

Views

Author

Paul D. Hanna, Aug 20 2018

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 24*x^3/3! + 256*x^4/4! + 3840*x^5/5! + 73024*x^6/6! + 1688064*x^7/7! + 45991936*x^8/8! + 1443102720*x^9/9! + 51249316864*x^10/10! + ...
such that cosh(x + A(x)) + sinh(x - A(x)) = 1.
RELATED SERIES.
(1) exp(A(x)) = 1 + x + 5*x^2/2! + 37*x^3/3! + 425*x^4/4! + 6601*x^5/5! + 129005*x^6/6! + 3044077*x^7/7! + 84239825*x^8/8! + ... + A318002(n)*x^n/n! + ...
which equals 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))).
(2) Let F(F(x)) = A(x) then
F(x) = x + 2*x^2/2! + 6*x^3/3! + 56*x^4/4! + 600*x^5/5! + 8432*x^6/6! + 144816*x^7/7! + 2892416*x^8/8! + 66721920*x^9/9! + ... + A318001(n)*x^n/n! + ...
where cosh(F(x) - F(-x)) - sinh(F(x) + F(-x)) = 1.

Links

Crossrefs

Cf. A318001 (A(A(x))), A318002 (exp(A(x))), A318005 (variant).

Programs

  • PARI
    {a(n) = my(A = log( 2*cosh(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sinh(2*x +x^2*O(x^n)))) )); n!*polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2) 1 = Sum_{n>=0} ( x + (-1)^n*A(x) )^n/n!.
(3a) 1 = cosh(A(x) + x) - sinh(A(x) - x).
(3b) 1 = cosh(x)*exp(-A(x)) + sinh(x)*exp(A(x)).
(3c) 1 = exp(x)*cosh(A(x)) - exp(-x)*sinh(A(x)).
(4a) A(x) = log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).
(4b) A(x) = log( (1 - sqrt(1 - 2*sinh(2*x))) / (2*sinh(x)) ).
(5) A(x) = F(F(x)) where F(x) is the e.g.f. of A318001, which satisfies: 1 = cosh(F(x) - F(-x)) - sinh(F(x) + F(-x)).
a(n) ~ 5^(1/4) * 2^(n - 1/2) * n^(n-1) / (exp(n) * log((1 + sqrt(5))/2)^(n - 1/2)). - Vaclav Kotesovec, Aug 21 2018

A318006 E.g.f. A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/( cos(A(-x)) + sin(A(-x)) ).

Original entry on oeis.org

1, 2, 6, 40, 360, 4592, 70896, 1279360, 26497920, 619457792, 16166151936, 466022394880, 14708199367680, 504453778491392, 18681868054910976, 742996971891097600, 31583887537425776640, 1429076863804079931392, 68575244394079858262016, 3478457209493103235563520, 185971933231431479545036800
Offset: 1

Views

Author

Paul D. Hanna, Aug 27 2018

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 360*x^5/5! + 4592*x^6/6! + 70896*x^7/7! + 1279360*x^8/8! + 26497920*x^9/9! + 619457792*x^10/10! + ...
such that
cos(A(x)) + sin(A(x)) = 1/( cos(A(-x)) + sin(A(-x)) )
and
sin(2*A(x)) = 2*sin(2*x)/(2 - sin(2*x)) = 2*sin(A(x))*cos(A(x)).
RELATED SERIES.
(a) A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 48064*x^6/6! + 989184*x^7/7! + 24218624*x^8/8! + ... + A318005(n)*x^n/n! + ...
where A(A(x)) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) ) /2.
(b) cos(A(x)) + sin(A(x)) = 1/(cos(A(-x)) - sin(A(-x))) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! - 59*x^5/5! - 239*x^6/6! - 421*x^7/7! + 4913*x^8/8! + 108361*x^9/9! + 1000321*x^10/10! + ...
where cos(A(x)) + sin(A(x)) = sqrt( (2 + sin(2*x))/(2 - sin(2*x)) ).

Links

Crossrefs

Cf. A318005, A318001, A200560.

Programs

  • Mathematica
    nmax = 25; Rest[CoefficientList[Series[ArcSin[4/(2 - Sin[2*x]) - 2]/2, {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Aug 28 2018 *)
  • PARI
    /* A(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) )/2 */
{a(n) = my(A = asin( 2*sin(2*x +x*O(x^n))/(2 - sin(2*x +x*O(x^n))) )/2 ); n!*polcoeff(A,n)}
for(n=1,25, print1(a(n),", "))
  • PARI
    /* From 1 = cos(A(A(x)) + x) + sin(A(A(x)) - x) */
{a(n) = my(A=[1,1]); for(i=1, n, A = concat(A,0);
A[#A] = -Vec(cos(subst(x*Ser(A),x,x*Ser(A)) + x) + sin(subst(x*Ser(A),x,x*Ser(A)) - x))[#A+1]/2; ); n!*A[n]}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2a) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(x) - (-1)^n*A(-x) )^n/n!.
(2b) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(A(x)) + (-1)^n*x )^n/n!.
(3a) 1 = cos( A(x) - A(-x) ) + sin( A(x) + A(-x) ).
(3b) 1 = ( cos(A(x)) + sin(A(x)) ) * ( cos(A(-x)) + sin(A(-x)) ).
(4a) 1 = cos(A(A(x)) + x) + sin(A(A(x)) - x).
(4b) 1 = ( cos(A(A(x))) + sin(A(A(x))) ) * (cos(x) - sin(x)).
(5a) A(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) )/2.
(5b) A(A(x)) = arcsin( sin(2*x)/(1 - sin(2*x)) )/2, which is the e.g.f. of A318005.
(6) cos(A(x)) + sin(A(x)) = sqrt( (2 + sin(2*x))/(2 - sin(2*x)) ).
a(n) ~ sqrt(3) * 5^(1/4) * 2^(n-2) * n^(n-1) / (exp(n) * (arcsin(2/3))^(n - 1/2)). - Vaclav Kotesovec, Aug 28 2018
Showing 1-2 of 2 results.

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