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A318004 E.g.f.: 2*cos(x) / (1 + sqrt(1 - 2*sin(2*x))). 1
1, 1, 3, 23, 249, 3601, 65163, 1420103, 36240369, 1060638241, 35030837523, 1289122462583, 52311218246889, 2320745189970481, 111753587921091483, 5805372695984119463, 323619830261141155809, 19269740737912396000321, 1220661620760214878827043, 81966845938603736180310743, 5815923258824904181135143129 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

FORMULA

E.g.f. A(x) satisfies:

(1a) A(x) = cos(x) + sin(x)*A(x)^2.

(1b) A(x) = cos(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sin(2*x)^n/2^n.

(2a) A(x) = (1 - sqrt(1 - 2*sin(2*x))) / (2*sin(x)).

(2b) A(x) = 2*cos(x) / (1 + sqrt(1 - 2*sin(2*x))).

a(n) ~ (1 + sqrt(3)) * 2^(2*n - 3/2) * 3^(n - 1/4) * n^(n-1) / (exp(n) * Pi^(n - 1/2)). - Vaclav Kotesovec, Aug 21 2018

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 249*x^4/4! + 3601*x^5/5! + 65163*x^6/6! + 1420103*x^7/7! + 36240369*x^8/8! + 1060638241*x^9/9! + ...

such that

A(x) = cos(x) + sin(x)*A(x)^2.

RELATED SERIES.

log(A(x)) = x + 2*x^2/2! + 16*x^3/3! + 160*x^4/4! + 2240*x^5/5! + 39392*x^6/6! + 841216*x^7/7! + 21130240*x^8/8! + 610734080*x^9/9! + ...

A(x)^2 = 1 + 2*x + 8*x^2/2! + 64*x^3/3! + 736*x^4/4! + 11072*x^5/5! + 206528*x^6/6! + 4607104*x^7/7! + 119766016*x^8/8! + ...

PROG

(PARI) {a(n) = my(A = 2*cos(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sin(2*x +x^2*O(x^n)))) ); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A318007.

Sequence in context: A122009 A151433 A199750 * A098681 A118790 A159017

Adjacent sequences:  A318001 A318002 A318003 * A318005 A318006 A318007

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 21 2018

STATUS

approved

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Last modified May 20 02:47 EDT 2022. Contains 353850 sequences. (Running on oeis4.)