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.

A281429 E.g.f.: C(x) + S(x) = exp( Integral C(x)^4 dx ) where C(x) and S(x) is described by A281428 and A281427, respectively.

Original entry on oeis.org

1, 1, 1, 5, 17, 145, 865, 10325, 88865, 1357825, 15335425, 284963525, 3993275825, 87274812625, 1462392957025, 36716097543125, 716611617346625, 20309401097610625, 452780458211706625, 14290053364475013125, 358439197464543820625, 12462411363013047060625
Offset: 0

Views

Author

Paul D. Hanna, Feb 01 2017

Keywords

Examples

			E.g.f: C(x) + S(x) = 1 + x + x^2/2! + 5*x^3/3! + 17*x^4/4! + 145*x^5/5! + 865*x^6/6! + 10325*x^7/7! + 88865*x^8/8! + 1357825*x^9/9! + 15335425*x^10/10! + 284963525*x^11/11! + 3993275825*x^12/12! + 87274812625*x^13/13! + 1462392957025*x^14/14! + 36716097543125*x^15/15! + 716611617346625*x^16/16! + 20309401097610625*x^17/17! + 452780458211706625*x^18/18! + 14290053364475013125*x^19/19! + 358439197464543820625*x^20/20! +...
where log( C(x) + S(x) ) = Integral C(x)^4 dx, and
C(x)^4 = 1 + 4*x^2/2! + 104*x^4/4! + 6880*x^6/6! + 855680*x^8/8! + 171673600*x^10/10! + 50628300800*x^12/12! + 20616410214400*x^14/14! + 11081874771968000*x^16/16! + 7600553402810368000*x^18/18! + 6477130108444835840000*x^20/20! +...

Crossrefs

Cf. A281427, A281428.

Programs

  • PARI
    {a(n) = my(S=x, C=1); for(i=0, n, S = intformal( C^5 +x*O(x^n)); C = 1 + intformal( S*C^4 ) ); n!*polcoeff(C+S, n)}
for(n=0, 30, print1(a(n), ", "))

A281428 E.g.f. C(x) satisfies: C(x) = cosh( Integral C(x)^4 dx ).

Original entry on oeis.org

1, 1, 17, 865, 88865, 15335425, 3993275825, 1462392957025, 716611617346625, 452780458211706625, 358439197464543820625, 347486061804813737430625, 404905203733448633060470625, 558379985997479451541000890625, 899457522079287575519574474640625, 1673555570600439849976672764510390625, 3562028724551236205811767300836022890625
Offset: 0

Views

Author

Paul D. Hanna, Jan 21 2017

Keywords

Examples

			E.g.f.: C(x) = 1 + x^2/2! + 17*x^4/4! + 865*x^6/6! + 88865*x^8/8! + 15335425*x^10/10! + 3993275825*x^12/12! + 1462392957025*x^14/14! + 716611617346625*x^16/16! + 452780458211706625*x^18/18! + 358439197464543820625*x^20/20! +...
such that
(1) C(x) = cosh( Integral C(x)^4 dx ),
(2) C(x)^2 - S(x)^2 = 1, and
(3) C(x) = 1 + Integral C(x)^4*S(x) dx,
where S(x) is described by A281427 and begins:
S(x) = x + 5*x^3/3! + 145*x^5/5! + 10325*x^7/7! + 1357825*x^9/9! + 284963525*x^11/11! + 87274812625*x^13/13! + 36716097543125*x^15/15! + 20309401097610625*x^17/17! + 14290053364475013125*x^19/19! +...

Programs

  • Mathematica
    a[n_] := Module[{S = x, C = 1, C5, SC4}, For[i = 0, i <= n, i++, C5 = C^5 + x*O[x]^(2n) // Normal; S = Integrate[C5 , x]; SC4 = S*C^4+O[x]^(2n) // Normal; C = 1 + Integrate[SC4, x]]; (2n)!*Coefficient[C, x, 2n]]; Array[a, 17, 0] (* Jean-François Alcover, Mar 01 2017, translated from Pari *)
  • PARI
    {a(n) = my(S=x, C=1); for(i=0, n, S = intformal( C^5 +x*O(x^(2*n))); C = 1 + intformal( S*C^4 ) ); (2*n)!*polcoeff(C, 2*n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. C(x) = ( d/dx Series_Reversion( x - x^3/3 ) )^(1/2).
E.g.f. C(x) = ( d/dx Series_Reversion( sin(x) - sin(x)^3/3 ) )^(1/3).
E.g.f. C(x) = ( d/dx Series_Reversion( sinh(x)*(2 + cosh(2*x))/(3*cosh(x)^3) ) )^(1/4).
E.g.f. C(x) = ( d/dx Series_Reversion( x*sqrt(1+x^2)*(3 + 2*x^2)/(3*(1 + x^2)^2) ) )^(1/5).
E.g.f. C(x) = d/dx Series_Reversion( Integral 1/G(x) dx ) where G(x) = e.g.f. of A281181.
E.g.f. C(x) = ( d/dx Series_Reversion( Integral (1 - x^2) dx ) )^(1/2).
E.g.f. C(x) = ( d/dx Series_Reversion( Integral cos(x)^3 dx ) )^(1/3).
E.g.f. C(x) = ( d/dx Series_Reversion( Integral 1/cosh(x)^4 dx ) )^(1/4).
E.g.f. C(x) = ( d/dx Series_Reversion( Integral 1/(1 + x^2)^(5/2) dx ) )^(1/5).
E.g.f. C(x) = ( d/dx Series_Reversion( Integral G(i*x)^6 dx ) )^(1/6) where G(x) = e.g.f. of A281181.
E.g.f. C(x) and related series S(x) (e.g.f. of A281427) satisfy:
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^5*S(x) dx.
Integrals.
(2.a) S(x) = Integral C(x)^5 dx.
(2.b) C(x) = 1 + Integral C(x)^4*S(x) dx.
Exponential.
(3.a) C(x) + S(x) = exp( Integral C(x)^4 dx ).
(3.b) C(x) = cosh( Integral C(x)^4 dx ).
(3.c) S(x) = sinh( Integral C(x)^4 dx ).
Derivatives.
(4.a) S'(x) = C(x)^5.
(4.b) C'(x) = C(x)^4*S(x).
(4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^4.
(4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^5*S(x).
Explicit Solutions.
(5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^(5/2) dx ).
(5.b) C(x)^1 = d/dx Series_Reversion( Integral 1/G(x) dx ) where G(x) = e.g.f. of A281181.
(5.c) C(x)^2 = d/dx Series_Reversion( Integral (1 - x^2) dx ).
(5.d) C(x)^3 = d/dx Series_Reversion( Integral cos(x)^3 dx ).
(5.e) C(x)^4 = d/dx Series_Reversion( Integral 1/cosh(x)^4 dx ).
(5.f) C(x)^5 = d/dx Series_Reversion( Integral 1/(1 + x^2)^(5/2) dx ).
(5.g) C(x)^6 = d/dx Series_Reversion( Integral G(i*x)^6 dx ) where G(x) = e.g.f. of A281181.
(5.h) C(x)^2 = d/dx Series_Reversion( x - x^3/3 ).
(5.j) C(x)^3 = d/dx Series_Reversion( sin(x) - sin(x)^3/3 ).
(5.j) C(x)^4 = d/dx Series_Reversion( sinh(x)*(2 + cosh(2*x))/(3*cosh(x)^3) ).
(5.k) C(x)^5 = d/dx Series_Reversion( x*sqrt(1+x^2)*(3 + 2*x^2)/(3*(1 + x^2)^2) ).
Showing 1-2 of 2 results.

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