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 1-5 of 5 results.

A280790 E.g.f. A(x) satisfies: A( sin( A( sinh(x) ) ) ) = x.

Original entry on oeis.org

1, 4, 2320, 9857600, 159122080000, 7098806416000000, 686863244097538560000, 143579312211740504320000000, 27634174819420517051458560000000, 103635121107833144489335056076800000000, -624322694794393812097710416148436992000000000, 9870191061692402402605200350045038131191808000000000, -258786046753018245774392957793266127246933652766720000000000, 11248188901093330352571154620038385487188031846809616384000000000000
Offset: 1

Views

Author

Paul D. Hanna, Jan 08 2017

Keywords

Comments

The series reversion of the e.g.f. is defined by A280792.

Examples

			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)! + ...

Links

Crossrefs

Cf. A280791, A280792, A280793, A318635.

Programs

  • 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),", "))

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)) ).

A280791 E.g.f. A(x) satisfies: A( tan( A( tanh(x) ) ) ) = x.

Original entry on oeis.org

1, 4, 400, 5364800, -367374176000, 143449000888960000, -181899009894595069440000, 627436681283593072503040000000, -5107564746905573153364013194240000000, 88171417366157389105207649269976371200000000, -2969272543655823399308577388625291953035264000000000, 182441297602875422577046590572630481727347923066880000000000
Offset: 1

Views

Author

Paul D. Hanna, Jan 08 2017

Keywords

Comments

The series reversion of the e.g.f. is defined by A280793.

Examples

			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)! +...

Links

Crossrefs

Cf. A280790, A280792, A280793.

Programs

  • 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),", "))

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)) ).

A280792 E.g.f. A(x) satisfies: A( arcsin( A( arcsinh(x) ) ) ) = x.

Original entry on oeis.org

1, -4, -304, 648896, 2650020096, -142483330376704, 24311838501965418496, -17572131142184492046434304, 31550058162566932127305417424896, -123841868587916789535717370523560443904, 969729634851676570691527174556498457233719296, -14068736567241332813708145418894026558391075423125504, 356436464229966658550949874743523835716465340767523041181696, -15023108679681039882374036580197265042861509571919315150655773999104
Offset: 1

Views

Author

Paul D. Hanna, Jan 09 2017

Keywords

Comments

The series reversion of the e.g.f. is defined by A280790.

Examples

			E.g.f.: 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! - 17572131142184492046434304*x^29/29! + 31550058162566932127305417424896*x^33/33! - 123841868587916789535717370523560443904*x^37/37! + 969729634851676570691527174556498457233719296*x^41/41! + ...
such that A( arcsin( A( arcsinh(x) ) ) ) = x.
Note that A( A( arcsin( arcsinh(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 - 19/22680*x^9 + 10139/97297200*x^13 + 3450547/463134672000*x^17 - 139143877321/49893498214560000*x^21 + 5935507446768901/3786916514485104000000*x^25 - 4413653374109964767/2220816151494708768000000*x^29 + ...
RELATED SERIES.
A( arcsinh(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! + ...
The series reversion of A( arcsinh(x) ) equals A( arcsin(x) ), which begins:
A( arcsin(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! + ... + A318635(n)*x^(2*n-1)/(2*n-1)! + ...
arcsinh( A(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 arcsinh( A(x) ) equals arcsin( A(x) ), which begins:
arcsin( A(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! + ...
The series reversion of A(x) begins:
Series_Reversion( 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! +...+ A280790(n)*x^(4*n-3)/(4*n-3)! +...

Links

Crossrefs

Cf. A280790, A280791, A280793, A318635.

Programs

  • PARI
    {a(n) = my(A=x +x*O(x^(4*n+1))); for(i=1,2*n, A = A + (x - subst( asin(A) ,x, asinh(A) ) )/2; H=A ); (4*n-3)!*polcoeff(A,4*n-3)}
for(n=1,20,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3)/(4*n-3)! satisfies:
(1) A( arcsin( A( arcsinh(x) ) ) ) = x.
(2) A( arcsinh( A( arcsin(x) ) ) ) = x.
(3) arcsin( A( arcsinh( A(x) ) ) ) = x.
(4) arcsinh( A( arcsin( A(x) ) ) ) = x.
(5) A( arcsinh(A(x)) ) = sin(x).
(6) A( arcsin(A(x)) ) = sinh(x).
(7) Series_Reversion( A(x) ) = arcsin( A(arcsinh(x)) ) = arcsinh( A(arcsin(x)) ).

A279837 E.g.f. A(x) satisfies: A( tanh( A(x) ) ) = tan(x).

Original entry on oeis.org

1, 2, 20, 496, 23120, 1747360, 195269568, 30288321792, 6227935871232, 1639388975800832, 537520438716580864, 214739554795652526080, 102653241459277667225600, 57838071113129054500200448, 37921092324167375349735014400, 28616681138798042948070311264256, 24621851021674983535130840611749888, 23955560260216279396643234915721281536
Offset: 1

Views

Author

Paul D. Hanna, Jan 11 2017

Keywords

Comments

First negative term is a(65), the coefficient of x^129 in A(x).
Apart from signs, essentially the same terms as A279839.

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( tanh( A(x) ) ) = tan(x).
Note that A(A(x)) is NOT equal to tan(arctanh(x)) nor arctanh(tan(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. A279839, A280791, A280793, A279836.

Programs

  • PARI
    {a(n) = my(X = x +x*O(x^(2*n)),A=X); for(i=1, 2*n, A = A + (tan(X) - subst(A,x, tanh(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( tanh( A(x) ) ) = tan(x).
(2) A( arctan( A(x) ) ) = arctanh(x).
(3) arctan( A( tanh( A(x) ) ) ) = x.
(4) tanh( A( arctan( A(x) ) ) ) = x.
(5) A( tanh( A( arctan(x) ) ) ) = x.
(6) A( arctan( A( tanh(x) ) ) ) = x.
(7) Series_Reversion( A(x) ) = tanh( A( arctan(x) ) ) = arctan( A( tanh(x) ) ), and equals the e.g.f. of A279839.

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 1-5 of 5 results.

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