A281182 E.g.f. C(x) + S(x) = exp( Integral C(x)^3 dx ) where C(x) and S(x) are described by A281181 and A281180, respectively.

1, 1, 1, 4, 13, 88, 493, 4672, 37369, 454144, 4732249, 70084096, 901188997, 15728822272, 240798388357, 4836914249728, 85948640603761, 1952137912385536, 39504564917358001, 1000749157519458304, 22726779729476308093, 635146072839001735168, 15998009117983994065693, 488855521088102855606272, 13526765851190230940840809, 448599416591747486039670784, 13528070218935445806530640649

Offset: 0

Author

Paul D. Hanna, Jan 16 2017

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 88*x^5/5! + 493*x^6/6! + 4672*x^7/7! + 37369*x^8/8! + 454144*x^9/9! + 4732249*x^10/10! + 70084096*x^11/11! + 901188997*x^12/12! +...
where A(x) = C(x) + S(x) and the series for C(x) and S(x) begin:
C(x) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 37369*x^8/8! + 4732249*x^10/10! + 901188997*x^12/12! + 240798388357*x^14/14! + 85948640603761*x^16/16! +...+ A281181(n)*x^(2*n)/(2*n)! +...
S(x) = S(x) = x + 4*x^3/3! + 88*x^5/5! + 4672*x^7/7! + 454144*x^9/9! + 70084096*x^11/11! + 15728822272*x^13/13! + 4836914249728*x^15/15! + 1952137912385536*x^17/17! +...+ A281180(n)*x^(2*n-1)/(2*n-1)! +...
such that C(s) + S(x) = exp( Integral C(x)^3 dx ).
The logarithm of the e.g.f. begins:
log(C(x) + S(x)) = x + 3*x^3/3! + 57*x^5/5! + 2739*x^7/7! + 246801*x^9/9! + 35822307*x^11/11! + 7636142793*x^13/13! + 2246286827091*x^15/15! +...
which equals Integral C(x)^3 dx.
Also, log(C(x) + S(x)) = Series_Reversion( Integral 1/cosh(x)^3 dx ).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A281180 (S), A281181 (C), A281183 (C^2), A281184 (C^3).

Programs

Mathematica

CoefficientList[Exp[InverseSeries[Series[(Sinh[x]/Cosh[x]^2 + ArcTan[Sinh[x]])/2, {x, 0, 30}], x]], x] * Range[0, 30]! (* Vaclav Kotesovec, Sep 02 2017 *)

PARI

{a(n) = my(S=x, C=1); for(i=1, n, S = intformal( C^4 +x*O(x^n)); C = 1 + intformal( S*C^3 ) ); n!*polcoeff(C + S, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

/* From S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ) */
{a(n) = my(S=x); S = serreverse( intformal( 1/(1 + x^2 +x*O(x^n))^2)); n!*polcoeff(sqrt(1+S^2) + S, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).

E.g.f. exp( Series_Reversion( ( sinh(x)/cosh(x)^2 + atan(sinh(x)) )/2 ) ).

E.g.f. C(x) + S(x) where related series S(x) and C(x) satisfy:

(1.a) C(x)^2 - S(x)^2 = 1.

(1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.

Integrals.

(2.a) S(x) = Integral C(x)^4 dx.

(2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.

Exponential.

(3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).

(3.b) C(x) = cosh( Integral C(x)^3 dx ).

(3.c) S(x) = sinh( Integral C(x)^3 dx ).

Derivatives.

(4.a) S'(x) = C(x)^4.

(4.b) C'(x) = C(x)^3*S(x).

(4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.

(4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).

Explicit Solutions.

(5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).

(5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).

(5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).

(5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).

(5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).