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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. 5
1, 3, 57, 2739, 246801, 35822307, 7636142793, 2246286827091, 871869519033249, 431649452286233283, 265466419357802436057, 198541440131880248161779, 177448471205103040365902001, 186781461066456338787698757027, 228695537454759099917373077023593, 322272887805877963568678968978370451, 517868815187736150011294497645677002049 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 21 02:48 EDT 2019. Contains 323434 sequences. (Running on oeis4.)