A318006 -id:A318006 - cp's OEIS frontend

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

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

Offset: 1

Paul D. Hanna, Aug 27 2018

nonn

			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.

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A318006, A318000, A200560.

{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),", "))

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

Paul D. Hanna, Aug 20 2018

sign

First negative term is a(27).

			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))) ).

Paul D. Hanna, Table of n, a(n) for n = 1..100

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

{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),", "))

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.

cp's OEIS Frontend

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

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