[go: up one dir, main page]

login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A279836
E.g.f. A(x) satisfies: A( sin( A(x) ) ) = sinh(x).
3
1, 1, 5, 113, 4505, 324545, 34312317, 5171466801, 1036525185393, 268061777199361, 86654517306871861, 34236056076864607345, 16224034929841344607625, 9077085568599515191480769, 5918716657866577845713460525, 4447229534037550877037585953073, 3813957492790787345317821024498657, 3702048025219670721125627874960351233
OFFSET
1,3
COMMENTS
First negative term is a(75), the coefficient of x^149 in A(x).
Apart from signs, essentially the same terms as A279838.
LINKS
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), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jan 11 2017
STATUS
approved