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%I #15 Feb 15 2019 09:55:24
%S 1,1,5,113,4505,324545,34312317,5171466801,1036525185393,
%T 268061777199361,86654517306871861,34236056076864607345,
%U 16224034929841344607625,9077085568599515191480769,5918716657866577845713460525,4447229534037550877037585953073,3813957492790787345317821024498657,3702048025219670721125627874960351233
%N E.g.f. A(x) satisfies: A( sin( A(x) ) ) = sinh(x).
%C First negative term is a(75), the coefficient of x^149 in A(x).
%C Apart from signs, essentially the same terms as A279838.
%H Paul D. Hanna, <a href="/A279836/b279836.txt">Table of n, a(n) for n = 1..100</a>
%F E.g.f. A(x) satisfies:
%F (1) A( sin( A(x) ) ) = sinh(x).
%F (2) A( arcsinh( A(x) ) ) = arcsin(x).
%F (3) arcsinh( A( sin( A(x) ) ) ) = x.
%F (4) sin( A( arcsinh( A(x) ) ) ) = x.
%F (5) A( sin( A( arcsinh(x) ) ) ) = x.
%F (6) A( arcsinh( A( sin(x) ) ) ) = x.
%F (7) Series_Reversion( A(x) ) = sin( A( arcsinh(x) ) ) = arcsinh( A( sin(x) ) ), and equals the e.g.f. of A279838.
%e 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! + ...
%e such that A( sin( A(x) ) ) = sinh(x).
%e 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.
%e The e.g.f. as a series with reduced fractional coefficients begins:
%e 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 + ...
%e RELATED SERIES.
%e 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! + ...
%e The series reversion of A( sin(x) ) equals A( arcsinh(x) ), which begins:
%e 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! + ...
%e 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! + ...
%e The series reversion of sin( A(x) ) equals arcsinh( A(x) ), which begins:
%e 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! + ...
%e The series reversion of A(x) = sin(A(arcsinh(x))) = arcsinh(A(sin(x))), and begins:
%e 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! + ...
%o (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)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A279838, A280790, A280792, A279837.
%K sign
%O 1,3
%A _Paul D. Hanna_, Jan 11 2017
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