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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 Paul D. Hanna, Table of n, a(n) for n = 0..200 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 A041433 A222764 Adjacent sequences:  A281179 A281180 A281181 * A281183 A281184 A281185 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 16 2017 STATUS approved

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Last modified May 24 06:53 EDT 2019. Contains 323529 sequences. (Running on oeis4.)