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 1-3 of 3 results.

A318005 E.g.f.: A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/(cos(x) - sin(x)).

Original entry on oeis.org

1, 4, 24, 224, 2880, 48064, 989184, 24218624, 687083520, 22151148544, 799546834944, 31934834253824, 1398132497448960, 66573473015578624, 3425078687463112704, 189331392774496845824, 11190654534195295027200, 704262689221037166690304, 47015904809670036594622464, 3318579148264602406039322624
Offset: 1

Views

Author

Paul D. Hanna, Aug 27 2018

Keywords

Examples

			E.g.f.: 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! + 687083520*x^9/9! + 22151148544*x^10/10! + ...
such that:
cos(A(x)) + sin(A(x)) = 1/( cos(x) - sin(x) ).
RELATED SERIES.
(a) cos(A(x)) + sin(A(x)) = 1/(cos(x) - sin(x)) = 1 + x + 3*x^2/2! + 11*x^3/3! + 57*x^4/4! + 361*x^5/5! + 2763*x^6/6! + ... + A001586(n)*x^n/n! + ...
(b) If F(F(x)) = A(x), then
F(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! + ... + A318006(n)*x^n/n! + ...
where F(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) ) /2.

Links

Crossrefs

Cf. A318006, A318000, A200560.

Programs

  • PARI
    {a(n) = my(A = asin( sin(2*x +x*O(x^n))/(1 - sin(2*x +x*O(x^n))) )/2 ); n!*polcoeff(A,n)}
for(n=1,20, print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(x) + (-1)^n*x )^n/n!.
(3a) 1 = cos(A(x) + x) + sin(A(x) - x).
(3b) 1 = ( cos(A(x)) + sin(A(x)) ) * ( cos(x) - sin(x) ).
(4) A(x) = arcsin( sin(2*x)/(1 - sin(2*x)) )/2.
a(n) = 2^(n-1) * A200560(n).

A318001 E.g.f. A(x) satisfies: cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)) = 1.

Original entry on oeis.org

1, 2, 6, 56, 600, 8432, 144816, 2892416, 66721920, 1732489472, 50144683776, 1604936139776, 56236356234240, 2137961925773312, 87642967518836736, 3863105286629851136, 182345733925971394560, 9130908475775186173952, 481864839159167717277696, 27108466364634568866922496, 1642481780780610712999034880
Offset: 1

Views

Author

Paul D. Hanna, Aug 20 2018

Keywords

Comments

First negative term is a(27).

Examples

			E.g.f.: A(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! + 1732489472*x^10/10! + 50144683776*x^11/11! + 1604936139776*x^12/12! + 56236356234240*x^13/13! + 2137961925773312*x^14/14! + 87642967518836736*x^15/15! + ...
such that cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)) = 1.
RELATED SERIES.
(1) exp(A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 105*x^4/4! + 1141*x^5/5! + 16083*x^6/6! + 276193*x^7/7! + 5561265*x^8/8! + 128834761*x^9/9! + 3365571363*x^10/10! + ...
which equals (sqrt(1 + 2*sinh(2*A(-x))) - 1) / (2*sinh(A(-x))).
(2) A(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! + ... + A318000(n)*x^n/n! + ...
which equals log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).

Links

Crossrefs

Cf. A318000 (A(A(x))), A318006 (variant).

Programs

  • PARI
    {a(n) = my(A=x+x^2 +x*O(x^n),S=x); for(i=1,n, S = (A - subst(A,x,-x))/2;
A = S + log(cosh(2*S) - 1 + sqrt(1 + (cosh(2*S) - 1)^2))/2;
A = (A - subst(serreverse(A),x,-x))/2 ); n!*polcoeff(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)^n * ( A(x) - (-1)^n*A(-x) )^n/n!.
(2b) 1 = Sum_{n>=0} ( x + (-1)^n*A(A(x)) )^n/n!.
(3a) 1 = cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)).
(3b) 1 = cosh(A(-x))*exp(-A(x)) - sinh(A(-x))*exp(A(x)).
(3c) 1 = cosh(x)*exp(-A(A(x))) + sinh(x)*exp(A(A(x))).
(4a) A(x) = log( 2*cosh(A(-x)) / (1 + sqrt(1 + 2*sinh(2*A(-x)))) ).
(4b) A(x) = log( (sqrt(1 + 2*sinh(2*A(-x))) - 1) / (2*sinh(A(-x))) ).
(5) A(A(x)) = log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ), which is the e.g.f. of A318000.

A318002 E.g.f.: 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))).

Original entry on oeis.org

1, 1, 5, 37, 425, 6601, 129005, 3044077, 84239825, 2675886481, 95979282005, 3837251617717, 169216980911225, 8160026826620761, 427179965967027005, 24127907244206776957, 1462542541799076574625, 94704025153744512625441, 6524332029969395884644005, 476487260493293293849001797, 36772596077297424381362590025, 2990260766874609440239439756521
Offset: 0

Views

Author

Paul D. Hanna, Aug 20 2018

Keywords

Examples

			E.g.f.: 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! + 2675886481*x^9/9! + ...
such that
A(x) = cosh(x) + sinh(x)*A(x)^2.
RELATED SERIES.
log(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! + ... + A318000(n)*x^n/n! + ...
where A( -log(A(x)) ) = exp(-x).
A(x)^2 = 1 + 2*x + 12*x^2/2! + 104*x^3/3! + 1296*x^4/4! + 21152*x^5/5! + 428352*x^6/6! + 10381184*x^7/7! + 293304576*x^8/8! + 9472819712*x^9/9! + ...

Crossrefs

Cf. A318000 (log(A(x))).

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(2 Cosh[x])/(1+Sqrt[1-2Sinh[2x]]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 31 2024 *)
  • PARI
    {a(n) = my(A = 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=0, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1a) A(x) = cosh(x) + sinh(x)*A(x)^2.
(1b) A(x) = cosh(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sinh(2*x)^n/2^n.
(1c) A(x) = (1 - sqrt(1 - 2*sinh(2*x))) / (2*sinh(x)).
(2) A( -log(A(x)) ) = exp(-x).
(3a) 1 = cosh(x + log(A(x))) + sinh(x - log(A(x))).
(3b) 1 = Sum_{n>=0} ( x + (-1)^n*log(A(x)) )^n/n!.
a(n) ~ 5^(1/4) * phi^(3/2) * 2^(n - 1/2) * n^(n-1) / (exp(n) * log(phi)^(n - 1/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 21 2018
Showing 1-3 of 3 results.

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