cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A279836 E.g.f. A(x) satisfies: A( sin( A(x) ) ) = sinh(x).

Original entry on oeis.org

1, 1, 5, 113, 4505, 324545, 34312317, 5171466801, 1036525185393, 268061777199361, 86654517306871861, 34236056076864607345, 16224034929841344607625, 9077085568599515191480769, 5918716657866577845713460525, 4447229534037550877037585953073, 3813957492790787345317821024498657, 3702048025219670721125627874960351233
Offset: 1

Views

Author

Paul D. Hanna, Jan 11 2017

Keywords

Comments

First negative term is a(75), the coefficient of x^149 in A(x).
Apart from signs, essentially the same terms as A279838.

Examples

			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! + ...

Links

Crossrefs

Cf. A279838, A280790, A280792, A279837.

Programs

  • 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), ", "))

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.

A279839 E.g.f. A(x) satisfies: A( tan( A(x) ) ) = tanh(x).

Original entry on oeis.org

1, -2, 20, -496, 23120, -1747360, 195269568, -30288321792, 6227935871232, -1639388975800832, 537520438716580864, -214739554795652526080, 102653241459277667225600, -57838071113129054500200448, 37921092324167375349735014400, -28616681138798042948070311264256, 24621851021674983535130840611749888
Offset: 1

Views

Author

Paul D. Hanna, Jan 11 2017

Keywords

Comments

Apart from signs, essentially the same terms as A279837.

Examples

			E.g.f.: A(x) = x - 2*x^3/3! + 20*x^5/5! - 496*x^7/7! + 23120*x^9/9! - 1747360*x^11/11! + 195269568*x^13/13! - 30288321792*x^15/15! + 6227935871232*x^17/17! - 1639388975800832*x^19/19! + 537520438716580864*x^21/21! - 214739554795652526080*x^23/23! + 102653241459277667225600*x^25/25! +...
such that A( tan( A(x) ) ) = tanh(x).
Note that A(A(x)) is NOT equal to tanh(atan(x)) nor atan(tanh(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 - 1/3*x^3 + 1/6*x^5 - 31/315*x^7 + 289/4536*x^9 - 10921/249480*x^11 + 78233/2494800*x^13 - 4381991/189189000*x^15 +...

Links

Crossrefs

Cf. A279837, A280791, A280793, A279838.

Programs

  • PARI
    {a(n) = my(X = x +x*O(x^(2*n)),A=X); for(i=1, 2*n, A = A + (tanh(X) - subst(A,x, tan(A) ) )/2; H=A ); (2*n-1)!*polcoeff(A, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A( tan( A(x) ) ) = tanh(x).
(2) A( atanh( A(x) ) ) = atan(x).
(3) atanh( A( tan( A(x) ) ) ) = x.
(4) tan( A( atanh( A(x) ) ) ) = x.
(5) A( tan( A( atanh(x) ) ) ) = x.
(6) A( atanh( A( tan(x) ) ) ) = x.
(7) Series_Reversion( A(x) ) = tan( A( atanh(x) ) ) = atanh( A( tan(x) ) ), and equals the e.g.f. of A279837.
Showing 1-2 of 2 results.

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