This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 15 04:23 EST 2019. Contains 329991 sequences. (Running on oeis4.)