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A280793 E.g.f. A(x) satisfies: A( arctan( A( arctanh(x) ) ) ) = x. 6
1, -4, 1616, -10233664, 605781862656, -195074044306023424, 226963189334487889924096, -745095268828143694162593398784, 5876637899238904537105181354518183936, -99252790021186158091252679600581668608671744, 3289325814605557759161838756845047127645003816370176, -199648823584758446510667095055905800597628128606583525474304 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The series reversion of the e.g.f. is defined by A280791.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..50

FORMULA

E.g.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3)/(4*n-3)! satisfies:

(1) A( arctan( A( arctanh(x) ) ) ) = x.

(2) A( arctanh( A( arctan(x) ) ) ) = x.

(3) arctan( A( arctanh( A(x) ) ) ) = x.

(4) arctanh( A( arctan( A(x) ) ) ) = x.

(5) A( arctanh(A(x)) ) = tan(x).

(6) A( arctan(A(x)) ) = tanh(x).

(7) Series_Reversion( A(x) ) = arctan( A(arctanh(x)) ) = arctanh( A(arctan(x)) ).

EXAMPLE

E.g.f.: A(x) = x - 4*x^5/5! + 1616*x^9/9! - 10233664*x^13/13! + 605781862656*x^17/17! - 195074044306023424*x^21/21! + 226963189334487889924096*x^25/25! - 745095268828143694162593398784*x^29/29! + 5876637899238904537105181354518183936*x^33/33! - 99252790021186158091252679600581668608671744*x^37/37! + 3289325814605557759161838756845047127645003816370176*x^41/41! + ...

such that A( arctan( A( arctanh(x) ) ) ) = x.

Note that A( A( arctan( arctanh(x) ) ) ) is NOT equal to x; the composition of these functions is not commutative.

The e.g.f. as a series with reduced fractional coefficients begins:

A(x) = x - 1/30*x^5 + 101/22680*x^9 - 22843/13899600*x^13 + 788778467/463134672000*x^17 - 190501996392601/49893498214560000*x^21 + 55410934896115207501/3786916514485104000000*x^25 - 15159002051353834923555367/179886108271071410208000000*x^29 + ...

RELATED SERIES.

A( arctanh(x) ) = x + 2*x^3/3! + 20*x^5/5! + 440*x^7/7! + 16400*x^9/9! + 944800*x^11/11! + 82388800*x^13/13! + 9583600000*x^15/15! + 1041175200000*x^17/17! + 136472188736000*x^19/19! + 168221708270720000*x^21/21! + 77192574087699200000*x^23/23! - 152078345729585600000000*x^25/25! + ...

The series reversion of A( arctanh(x) ) equals A( arctan(x) ), which begins:

A( arctan(x) ) = x - 2*x^3/3! + 20*x^5/5! - 440*x^7/7! + 16400*x^9/9! - 944800*x^11/11! + 82388800*x^13/13! - 9583600000*x^15/15! + ...

arctanh( A(x) ) = x + 2*x^3/3! + 20*x^5/5! + 552*x^7/7! + 29840*x^9/9! + 2520352*x^11/11! + 302768960*x^13/13! + 51218036352*x^15/15! + 12015036698880*x^17/17! + 3457794697175552*x^19/19! + 1042442536703513600*x^21/21! + 437297928076611069952*x^23/23! + 444983819928674567557120*x^25/25! + ...

The series reversion of arctanh( A(x) ) equals arctan( A(x) ), which begins:

arctan( A(x) ) = x - 2*x^3/3! + 20*x^5/5! - 552*x^7/7! + 29840*x^9/9! - 2520352*x^11/11! + 302768960*x^13/13! - 51218036352*x^15/15! + ...

The series reversion of A(x) begins:

Series_Reversion( A(x) ) = x + 4*x^5/5! + 400*x^9/9! + 5364800*x^13/13! - 367374176000*x^17/17! + 143449000888960000*x^21/21! - 181899009894595069440000*x^25/25! +...+ A280791(n)*x^(4*n-3)/(4*n-3)! + ...

PROG

(PARI) {a(n) = my(A=x +x*O(x^(4*n+1))); for(i=1, 2*n, A = A + (x - subst( atan(A) , x, atanh(A) ) )/2; ); (4*n-3)!*polcoeff(A, 4*n-3)}

for(n=1, 20, print1(a(n), ", "))

CROSSREFS

Cf. A280790, A280791, A280792.

Sequence in context: A278549 A229664 A265661 * A160225 A316484 A278794

Adjacent sequences:  A280790 A280791 A280792 * A280794 A280795 A280796

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jan 09 2017

STATUS

approved

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Last modified January 20 23:47 EST 2021. Contains 340332 sequences. (Running on oeis4.)