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 1-4 of 4 results.

A290879 E.g.f. L(x) of aerated sequence satisfies: L(x) = Integral 1 / sqrt( cosh(2*L(x)) ) dx.

Original entry on oeis.org

1, -2, 44, -2840, 367760, -79719200, 26016555200, -11921650083200, 7300922254496000, -5759532173685056000, 5688335131502291840000, -6875441991877610827520000, 9983390897443366347676160000, -17148197258942716354314368000000, 34392433372153876998446324480000000, -79646680171456811546338888517120000000, 210930739548407111241584046599398400000000
Offset: 1

Views

Author

Paul D. Hanna, Aug 13 2017

Keywords

Examples

			E.g.f.: L(x) = x - 2*x^3/3! + 44*x^5/5! - 2840*x^7/7! + 367760*x^9/9! - 79719200*x^11/11! + 26016555200*x^13/13! - 11921650083200*x^15/15! + 7300922254496000*x^17/17! - 5759532173685056000*x^19/19! + 5688335131502291840000*x^21/21! +...
where
1/sqrt( cosh(2*L(x)) ) = 1 - 2*x^2/2! + 44*x^4/4! - 2840*x^6/6! + 367760*x^8/8! - 79719200*x^10/10! + 26016555200*x^12/12! - 11921650083200*x^14/14! +...

Links

Crossrefs

Cf. A290880, A290881, A290882, A290883, A153302.

Programs

  • Maple
    Q:= series(Int(sqrt(cosh(2*t)),t),t,100):
S:= series(RootOf(Q-y,t),y,100):
seq(coeff(S,y,j)*j!,j=1..100,2); # Robert Israel, Aug 13 2017
  • Mathematica
    terms = 17; m = 2 terms; cc = CoefficientList[InverseSeries[Integrate[Sqrt[ Cosh[2 x]] + O[x]^m, x], x], x]; DeleteCases[cc * Range[0, m-1]!, 0] (* Jean-François Alcover, Apr 02 2019 *)
  • PARI
    {a(n) = my(L=x); for(i=1,n, L = intformal( 1/sqrt(cosh(2*L + O(x^(2*n+2)))) )); (2*n-1)!*polcoeff(L,2*n-1)}
for(n=1,20, print1(a(n),", "))
  • PARI
    {a(n) = my(L=x); L = serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) )); (2*n-1)!*polcoeff(L,2*n-1)}
for(n=1,20, print1(a(n),", "))

Formula

E.g.f.: L(x) = Series_Reversion( Integral sqrt( cosh(2*x) ) dx ).
Let C(x) and S(x) be the e.g.f.s of A290880 and A290881, respectively, then e.g.f. L(x) satisfies:
(1) L(x) = log(C(x) + S(x)),
(2) cosh(2*L(x)) = C(x)^2 + S(x)^2,
(3) cosh(L(x)) = C(x) and sinh(L(x)) = S(x),
where C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.

A290880 E.g.f. C(x) satisfies: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where S(x) is the e.g.f. of A290881.

Original entry on oeis.org

1, 1, -7, 265, -24175, 4037425, -1070526775, 412826556025, -218150106913375, 151297155973926625, -133288452772763494375, 145378048431548466795625, -192296944484564858674279375, 303266384253858232005535140625, -562167814015907092875287424484375, 1210147640238238850996978598797265625, -2993757681527630470101347134338702109375
Offset: 0

Views

Author

Paul D. Hanna, Aug 13 2017

Keywords

Examples

			E.g.f.: C(x) = 1 + x^2/2! - 7*x^4/4! + 265*x^6/6! - 24175*x^8/8! + 4037425*x^10/10! - 1070526775*x^12/12! + 412826556025*x^14/14! - 218150106913375*x^16/16! + 151297155973926625*x^18/18! - 133288452772763494375*x^20/20! +...
such that C(x)^2 - S(x)^2 = 1 where S(x) begins:
S(x) = x - x^3/3! + 25*x^5/5! - 1705*x^7/7! + 227665*x^9/9! - 50333425*x^11/11! + 16655398825*x^13/13! - 7711225809625*x^15/15! + 4760499335502625*x^17/17! - 3779764853639958625*x^19/19! + 3752942823715824285625*x^21/21! +...

Links

Crossrefs

Cf. A290879, A290881, A290882, A290883, A153302.

Programs

  • PARI
    {a(n) = my(C=1,S=x); for(i=1,n, C = 1 + intformal( S/sqrt(C^2 + S^2 + O(x^(2*n+2))) ); S = intformal( C/sqrt(C^2 + S^2)) ); (2*n)!*polcoeff(C,2*n)}
for(n=0,20, print1(a(n),", "))
  • PARI
    {a(n) = my(C=1); C = cosh( serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) ) )); (2*n)!*polcoeff(C,2*n)}
for(n=0,20, print1(a(n),", "))

Formula

E.g.f.: C(x) = cosh( Series_Reversion( Integral sqrt( cosh(2*x) ) dx ) ).
Let S(x) be the e.g.f. of A290881, then:
(1) C'(x) = S(x) / sqrt(C(x)^2 + S(x)^2).
(2) S'(x) = C(x) / sqrt(C(x)^2 + S(x)^2).
such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.

A290881 E.g.f. S(x) satisfies: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where C(x) is the e.g.f. of A290880.

Original entry on oeis.org

1, -1, 25, -1705, 227665, -50333425, 16655398825, -7711225809625, 4760499335502625, -3779764853639958625, 3752942823715824285625, -4556465805050372544735625, 6641455313355871353308640625, -11445605320939175012746492140625, 23021828780691053491298409381015625, -53450977127256739279274500814544765625
Offset: 1

Views

Author

Paul D. Hanna, Aug 13 2017

Keywords

Examples

			E.g.f.: S(x) = x - x^3/3! + 25*x^5/5! - 1705*x^7/7! + 227665*x^9/9! - 50333425*x^11/11! + 16655398825*x^13/13! - 7711225809625*x^15/15! + 4760499335502625*x^17/17! - 3779764853639958625*x^19/19! + 3752942823715824285625*x^21/21! +...
such that C(x)^2 - S(x)^2 = 1 where C(x) begins:
C(x) = 1 + x^2/2! - 7*x^4/4! + 265*x^6/6! - 24175*x^8/8! + 4037425*x^10/10! - 1070526775*x^12/12! + 412826556025*x^14/14! - 218150106913375*x^16/16! + 151297155973926625*x^18/18! - 133288452772763494375*x^20/20! +...

Links

Crossrefs

Cf. A290879, A290880, A290882, A290883, A153302.

Programs

  • PARI
    {a(n) = my(C=1,S=x); for(i=1,n, C = 1 + intformal( S/sqrt(C^2 + S^2 + O(x^(2*n+2))) ); S = intformal( C/sqrt(C^2 + S^2)) ); (2*n-1)!*polcoeff(S,2*n-1)}
for(n=1,20, print1(a(n),", "))
  • PARI
    {a(n) = my(C=1); S = serreverse( intformal( sqrt( (1+2*x^2) / (1+x^2 + O(x^(2*n+2))) ) )); (2*n-1)!*polcoeff(S,2*n-1)}
for(n=1,20, print1(a(n),", "))
  • PARI
    {a(n) = my(S=x); S = sinh( serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) ) )); (2*n-1)!*polcoeff(S,2*n-1)}
for(n=1,20, print1(a(n),", "))

Formula

E.g.f.: S(x) = Series_Reversion( Integral sqrt( (1 + 2*x^2) / (1 + x^2) ) dx ).
E.g.f.: S(x) = sinh( Series_Reversion( Integral sqrt( cosh(2*x) ) dx ) ).
Let C(x) be the e.g.f. of A290880, then:
(1) C'(x) = S(x) / sqrt(C(x)^2 + S(x)^2),
(2) S'(x) = C(x) / sqrt(C(x)^2 + S(x)^2),
such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.

A290882 E.g.f. E(x) = C(x) + S(x) such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where C(x) is the e.g.f. of A290880 and S(x) is the e.g.f. of A290881.

Original entry on oeis.org

1, 1, 1, -1, -7, 25, 265, -1705, -24175, 227665, 4037425, -50333425, -1070526775, 16655398825, 412826556025, -7711225809625, -218150106913375, 4760499335502625, 151297155973926625, -3779764853639958625, -133288452772763494375, 3752942823715824285625, 145378048431548466795625, -4556465805050372544735625, -192296944484564858674279375, 6641455313355871353308640625
Offset: 0

Views

Author

Paul D. Hanna, Aug 13 2017

Keywords

Examples

			E.g.f.: E(x) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! + 25*x^5/5! + 265*x^6/6! - 1705*x^7/7! - 24175*x^8/8! + 227665*x^9/9! + 4037425*x^10/10! - 50333425*x^11/11! - 1070526775*x^12/12! + 16655398825*x^13/13! + 412826556025*x^14/14! - 7711225809625*x^15/15! - 218150106913375*x^16/16! +...
such that E(x) = C(x) + S(x) where
S(x) = x - x^3/3! + 25*x^5/5! - 1705*x^7/7! + 227665*x^9/9! - 50333425*x^11/11! + 16655398825*x^13/13! - 7711225809625*x^15/15! + 4760499335502625*x^17/17! - 3779764853639958625*x^19/19! + 3752942823715824285625*x^21/21! +...
C(x) = 1 + x^2/2! - 7*x^4/4! + 265*x^6/6! - 24175*x^8/8! + 4037425*x^10/10! - 1070526775*x^12/12! + 412826556025*x^14/14! - 218150106913375*x^16/16! + 151297155973926625*x^18/18! - 133288452772763494375*x^20/20! +...
These series satisfy: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.

Links

Crossrefs

Cf. A290879, A290880, A290881, A290883.

Programs

  • PARI
    {a(n) = my(C=1,S=x); for(i=1,n, C = 1 + intformal( S/sqrt(C^2 + S^2 + O(x^(n+2))) ); S = intformal( C/sqrt(C^2 + S^2)) ); n!*polcoeff(C + S,n)}
for(n=0,30, print1(a(n),", "))
  • PARI
    {a(n) = my(E=1); E = exp( serreverse( intformal( sqrt(cosh(2*x + O(x^(n+2)))) ) )); n!*polcoeff(E,n)}
for(n=0,30, print1(a(n),", "))

Formula

E.g.f.: E(x) = exp( Series_Reversion( Integral sqrt( cosh(2*x) ) dx ) ).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.