A281184 E.g.f. C(x)^3 = d/dx log(C(x) + S(x)), where C(x) and S(x) are described by A281181 and A281180, respectively.

1, 3, 57, 2739, 246801, 35822307, 7636142793, 2246286827091, 871869519033249, 431649452286233283, 265466419357802436057, 198541440131880248161779, 177448471205103040365902001, 186781461066456338787698757027, 228695537454759099917373077023593, 322272887805877963568678968978370451, 517868815187736150011294497645677002049

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

E.g.f. C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).

E.g.f. C(x)^3 = d/dx Series_Reversion( ( sinh(x)/cosh(x)^2 + atan(sinh(x)) )/2 ).

E.g.f. C(x)^3 = d/dx log(C(x) + S(x)) where C(x) and S(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 ).

Example

E.g.f.: C(x)^3 = 1 + 3*x^2/2! + 57*x^4/4! + 2739*x^6/6! + 246801*x^8/8! + 35822307*x^10/10! + 7636142793*x^12/12! + 2246286827091*x^14/14! + 871869519033249*x^16/16! + 431649452286233283*x^18/18! +...
where related series 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)! +...
Also, the logarithm of C(x) + S(x) 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.

Prog

(PARI) {a(n) = my(S=x, C=1); for(i=1, n, S = intformal( C^4 +x*O(x^(2*n))); C = 1 + intformal( S*C^3 ) ); (2*n)!*polcoeff(C^3, 2*n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* E.g.f. d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ) */
{a(n) = my(C3=1); C3 = deriv( serreverse( intformal( 1/cosh(x +x*O(x^(2*n)))^3 ) ) ); (2*n)!*polcoeff(C3, 2*n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A281180 (S), A281181 (C), A281182 (C+S), A281183 (C^2).

Sequence in context: A012196 A012090 A210674 * A012064 A012204 A000281

Adjacent sequences: A281181 A281182 A281183 * A281185 A281186 A281187

Keyword

nonn

Author

Paul D. Hanna, Jan 17 2017

Status

approved