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

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

Offset

0,3

Links

Table of n, a(n) for n=0..16.

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 ) )^(1/6) 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) ).

Example

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! +...

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 *)

Prog

(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), ", "))

Crossrefs

Sequence in context: A173983 A231778 A355468 * A221324 A124235 A218660

Adjacent sequences: A281425 A281426 A281427 * A281429 A281430 A281431

Keyword

nonn

Author

Paul D. Hanna, Jan 21 2017

Status

approved