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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
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 ).
EXAMPLE
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 ).
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 *)
PROG
(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), ", "))
CROSSREFS
Cf. A281180 (S), A281181 (C), A281183 (C^2), A281184 (C^3).
Sequence in context: A012134 A088946 A131590 * A200850 A326565 A041433
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 16 2017
STATUS
approved

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)