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E.g.f. A(x) satisfies: A( sin( A( sinh(x) ) ) ) = x.
+10
7
1, 4, 2320, 9857600, 159122080000, 7098806416000000, 686863244097538560000, 143579312211740504320000000, 27634174819420517051458560000000, 103635121107833144489335056076800000000, -624322694794393812097710416148436992000000000, 9870191061692402402605200350045038131191808000000000, -258786046753018245774392957793266127246933652766720000000000, 11248188901093330352571154620038385487188031846809616384000000000000
COMMENTS
The series reversion of the e.g.f. is defined by A280792.
FORMULA
E.g.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3)/(4*n-3)! satisfies:
(1) A( sin( A( sinh(x) ) ) ) = x.
(2) A( sinh( A( sin(x) ) ) ) = x.
(3) sin( A( sinh( A(x) ) ) ) = x.
(4) sinh( A( sin( A(x) ) ) ) = x.
(5) A( sinh(A(x)) ) = asin(x).
(6) A( sin(A(x)) ) = asinh(x).
(7) Series_Reversion( A(x) ) = sin( A(sinh(x)) ) = sinh( A(sin(x)) ).
EXAMPLE
E.g.f.: A(x) = x + 4*x^5/5! + 2320*x^9/9! + 9857600*x^13/13! + 159122080000*x^17/17! + 7098806416000000*x^21/21! + 686863244097538560000*x^25/25! + 143579312211740504320000000*x^29/29! + 27634174819420517051458560000000*x^33/33! + 103635121107833144489335056076800000000*x^37/37! - 624322694794393812097710416148436992000000000*x^41/41! +...
such that A( sin( A( sinh(x) ) ) ) = x.
Note that A( A( sin( sinh(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 + 29/4536*x^9 + 6161/3891888*x^13 + 382505/855017856*x^17 + 50189525/361219896576*x^21 + 134894899309/3046287794457600*x^25 + 195216389950265/12021626449023916032*x^29 + ...
RELATED SERIES.
A( sinh(x) ) = x + x^3/3! + 5*x^5/5! + 141*x^7/7! + 6185*x^9/9! + 482681*x^11/11! + 55181165*x^13/13! + 8650849221*x^15/15! + 1806577140945*x^17/17! + 482615036315761*x^19/19! + 160833575943581525*x^21/21! + 65507016886932658301*x^23/23! + 32006289578900322278905*x^25/25! + ...
The series reversion of A( sinh(x) ) equals A( sin(x) ), which begins:
A( sin(x) ) = x - x^3/3! + 5*x^5/5! - 141*x^7/7! + 6185*x^9/9! - 482681*x^11/11! + 55181165*x^13/13! + ...
sinh( A(x) ) = x + x^3/3! + 5*x^5/5! + 85*x^7/7! + 2825*x^9/9! + 151625*x^11/11! + 12098125*x^13/13! + 1339476125*x^15/15! + 196410020625*x^17/17! + 37062144900625*x^19/19! + 8772471210303125*x^21/21! + 2519410212081953125*x^23/23! + 854580849916226265625*x^25/25! + ... + A318635(n)*x^(2*n-1)/(2*n-1)! + ... The series reversion of sinh( A(x) ) equals sin( A(x) ), which begins:
sin( A(x) ) = x - x^3/3! + 5*x^5/5! - 85*x^7/7! + 2825*x^9/9! - 151625*x^11/11! + 12098125*x^13/13! + ...
The series reversion of A(x) = sin(A(sinh(x))) = sinh(A(sin(x))), and begins:
Series_Reversion( A(x) ) = x - 4*x^5/5! - 304*x^9/9! + 648896*x^13/13! + 2650020096*x^17/17! - 142483330376704*x^21/21! + 24311838501965418496*x^25/25! +...+ A280792(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( sin(A) , x, sinh(A) ) )/2; H=A ); (4*n-3)!*polcoeff(A, 4*n-3)}
for(n=1, 20, print1(a(n), ", "))
E.g.f. A(x) satisfies: A( tan( A( tanh(x) ) ) ) = x.
+10
6
1, 4, 400, 5364800, -367374176000, 143449000888960000, -181899009894595069440000, 627436681283593072503040000000, -5107564746905573153364013194240000000, 88171417366157389105207649269976371200000000, -2969272543655823399308577388625291953035264000000000, 182441297602875422577046590572630481727347923066880000000000
COMMENTS
The series reversion of the e.g.f. is defined by A280793.
FORMULA
E.g.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3)/(4*n-3)! satisfies:
(1) A( tan( A( tanh(x) ) ) ) = x.
(2) A( tanh( A( tan(x) ) ) ) = x.
(3) tan( A( tanh( A(x) ) ) ) = x.
(4) tanh( A( tan( A(x) ) ) ) = x.
(5) A( tanh(A(x)) ) = arctan(x).
(6) A( tan(A(x)) ) = arctanh(x).
(7) Series_Reversion( A(x) ) = tan( A(tanh(x)) ) = tanh( A(tan(x)) ).
EXAMPLE
E.g.f.: 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! + 627436681283593072503040000000*x^29/29! - 5107564746905573153364013194240000000*x^33/33! + 88171417366157389105207649269976371200000000*x^37/37! - 2969272543655823399308577388625291953035264000000000*x^41/41! +...
such that A( tan( A( tanh(x) ) ) ) = x.
Note that A( A( tan( tanh(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 + 5/4536*x^9 + 479/555984*x^13 - 883111/855017856*x^17 + 1014203909/361219896576*x^21 - 5103375762413/435183970636800*x^25 + 77553540368447155/1092875131729446912*x^29 +...
RELATED SERIES.
A( tanh(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 A( tanh(x) ) equals A( tan(x) ), which begins:
A( tan(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! +...
tanh( A(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 tanh( A(x) ) equals tan( A(x) ), which begins:
tan( A(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! +...
The series reversion of A(x) = tan(A(tanh(x))) = tanh(A(tan(x))), and begins:
Series_Reversion( 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! +...+ A280793(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( tan(A) , x, tanh(A) ) )/2; ); (4*n-3)!*polcoeff(A, 4*n-3)}
for(n=1, 20, print1(a(n), ", "))
E.g.f. A(x) satisfies: A( arctan( A( arctanh(x) ) ) ) = x.
+10
6
1, -4, 1616, -10233664, 605781862656, -195074044306023424, 226963189334487889924096, -745095268828143694162593398784, 5876637899238904537105181354518183936, -99252790021186158091252679600581668608671744, 3289325814605557759161838756845047127645003816370176, -199648823584758446510667095055905800597628128606583525474304
COMMENTS
The series reversion of the e.g.f. is defined by A280791.
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), ", "))
E.g.f. A(x) satisfies: A( sin( A(x) ) ) = sinh(x).
+10
3
1, 1, 5, 113, 4505, 324545, 34312317, 5171466801, 1036525185393, 268061777199361, 86654517306871861, 34236056076864607345, 16224034929841344607625, 9077085568599515191480769, 5918716657866577845713460525, 4447229534037550877037585953073, 3813957492790787345317821024498657, 3702048025219670721125627874960351233
COMMENTS
First negative term is a(75), the coefficient of x^149 in A(x).
Apart from signs, essentially the same terms as A279838.
FORMULA
E.g.f. A(x) satisfies:
(1) A( sin( A(x) ) ) = sinh(x).
(2) A( arcsinh( A(x) ) ) = arcsin(x).
(3) arcsinh( A( sin( A(x) ) ) ) = x.
(4) sin( A( arcsinh( A(x) ) ) ) = x.
(5) A( sin( A( arcsinh(x) ) ) ) = x.
(6) A( arcsinh( A( sin(x) ) ) ) = x.
(7) Series_Reversion( A(x) ) = sin( A( arcsinh(x) ) ) = arcsinh( A( sin(x) ) ), and equals the e.g.f. of A279838.
EXAMPLE
E.g.f.: A(x) = x + x^3/3! + 5*x^5/5! + 113*x^7/7! + 4505*x^9/9! + 324545*x^11/11! + 34312317*x^13/13! + 5171466801*x^15/15! + 1036525185393*x^17/17! + 268061777199361*x^19/19! + 86654517306871861*x^21/21! + 34236056076864607345*x^23/23! + 16224034929841344607625*x^25/25! + ...
such that A( sin( A(x) ) ) = sinh(x).
Note that A(A(x)) is NOT equal to sinh(arcsin(x)) nor arcsin(sinh(x)) since the composition of these functions is not commutative.
The e.g.f. as a series with reduced fractional coefficients begins:
A(x) = x + (1/6)*x^3 + (1/24)*x^5 + (113/5040)*x^7 + (901/72576)*x^9 + (64909/7983360)*x^11 + (879803/159667200)*x^13 + (1723822267/435891456000)*x^15 + ...
RELATED SERIES.
A( sin(x) ) = x - 4*x^5/5! + 28*x^7/7! - 976*x^9/9! + 38016*x^11/11! - 3272736*x^13/13! + 321487680*x^15/15! - 47598285056*x^17/17! + 8350711540224*x^19/19! - 1819783398735872*x^21/21! + ...
The series reversion of A( sin(x) ) equals A( arcsinh(x) ), which begins:
A( arcsinh(x) ) = x + 4*x^5/5! - 28*x^7/7! + 2992*x^9/9! - 126720*x^11/11! + 20505952*x^13/13! - 2396136256*x^15/15! + ...
sin( A(x) ) = x - 4*x^5/5! - 28*x^7/7! - 976*x^9/9! - 38016*x^11/11! - 3272736*x^13/13! - 321487680*x^15/15! - 47598285056*x^17/17! - 8350711540224*x^19/19! - 1819783398735872*x^21/21! + ...
The series reversion of sin( A(x) ) equals arcsinh( A(x) ), which begins:
arcsinh( A(x) ) = x + 4*x^5/5! + 28*x^7/7! + 2992*x^9/9! + 126720*x^11/11! + 20505952*x^13/13! + 2396136256*x^15/15! + ...
The series reversion of A(x) = sin(A(arcsinh(x))) = arcsinh(A(sin(x))), and begins:
Series_Reversion( A(x) ) = x - x^3/3! + 5*x^5/5! - 113*x^7/7! + 4505*x^9/9! - 324545*x^11/11! + 34312317*x^13/13! - 5171466801*x^15/15! + ...
PROG
(PARI) {a(n) = my(X = x +x*O(x^(2*n)), A=X); for(i=1, 2*n, A = A + (sinh(X) - subst(A, x, sin(A) ) )/2; H=A ); (2*n-1)!*polcoeff(A, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))
E.g.f. A(x) satisfies: A( sinh( A(x) ) ) = sin(x).
+10
3
1, -1, 5, -113, 4505, -324545, 34312317, -5171466801, 1036525185393, -268061777199361, 86654517306871861, -34236056076864607345, 16224034929841344607625, -9077085568599515191480769, 5918716657866577845713460525, -4447229534037550877037585953073, 3813957492790787345317821024498657, -3702048025219670721125627874960351233
COMMENTS
Apart from signs, essentially the same terms as A279836.
FORMULA
E.g.f. A(x) satisfies:
(1) A( sinh( A(x) ) ) = sin(x).
(2) A( arcsin( A(x) ) ) = arcsinh(x).
(3) arcsin( A( sinh( A(x) ) ) ) = x.
(4) sinh( A( arcsin( A(x) ) ) ) = x.
(5) A( sinh( A( arcsin(x) ) ) ) = x.
(6) A( arcsin( A( sinh(x) ) ) ) = x.
(7) Series_Reversion( A(x) ) = sinh( A( arcsin(x) ) ) = arcsin( A( sinh(x) ) ), and equals the e.g.f. of A279836.
EXAMPLE
E.g.f.: A(x) = x - x^3/3! + 5*x^5/5! - 113*x^7/7! + 4505*x^9/9! - 324545*x^11/11! + 34312317*x^13/13! - 5171466801*x^15/15! + 1036525185393*x^17/17! - 268061777199361*x^19/19! + 86654517306871861*x^21/21! - 34236056076864607345*x^23/23! + 16224034929841344607625*x^25/25! + ...
such that A( sinh( A(x) ) ) = sin(x).
Note that A(A(x)) is NOT equal to sin(arcsinh(x)) nor arcsinh(sin(x)) since the composition of these functions is not commutative.
The e.g.f. as a series with reduced fractional coefficients begins:
A(x) = x - (1/6)*x^3 + (1/24)*x^5 - (113/5040)*x^7 + (901/72576)*x^9 - (64909/7983360)*x^11 + (879803/159667200)*x^13 - (1723822267/435891456000)*x^15 + ...
PROG
(PARI) {a(n) = my(X = x +x*O(x^(2*n)), A=X); for(i=1, 2*n, A = A + (sin(X) - subst(A, x, sinh(A) ) )/2; H=A ); (2*n-1)!*polcoeff(A, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))
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