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A281181 E.g.f. C(x) satisfies: C(x) = cosh( Integral C(x)^3 dx ).

%I A281181 #52 Mar 08 2025 06:02:56
%S A281181 1,1,13,493,37369,4732249,901188997,240798388357,85948640603761,
%T A281181 39504564917358001,22726779729476308093,15998009117983994065693,
%U A281181 13526765851190230940840809,13528070218935445806530640649,15795819619923464298050697616117,21294937666865806704402646632389557,32828500597549179599563478551377297121,57385924456400269824204023290894357442401,112904615348383588847189789579363784912180973
%N A281181 E.g.f. C(x) satisfies: C(x) = cosh( Integral C(x)^3 dx ).
%C A281181 From _Paul Curtz_, Jan 20 2017: (Start)
%C A281181 a(n) mod 10 = periodic sequence of length 8: repeat [1, 1, 3, 3, 9, 9, 7, 7] = duplicated A001148(n).
%C A281181 a(n) mod 9 = 1, followed by period 3: repeat [1, 4, 7]. See A100402. See also A281280, A281182, A281183, A281184 (1, followed by 3's).
%C A281181 a(n+p) - a(n) is a multiple of 12. (End)
%H A281181 Paul D. Hanna, <a href="/A281181/b281181.txt">Table of n, a(n) for n = 0..100</a>
%F A281181 E.g.f. C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
%F A281181 E.g.f. C(x) = d/dx Series_Reversion( ( x*sqrt(1 - x^2) + asin(x) )/2 ).
%F A281181 E.g.f. C(x) = ( d/dx Series_Reversion( Integral cos(x)^2 dx ) )^(1/2).
%F A281181 E.g.f. C(x) = ( d/dx Series_Reversion( (2*x + sin(2*x))/4 ) )^(1/2).
%F A281181 E.g.f. C(x) = ( d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ) )^(1/3).
%F A281181 E.g.f. C(x) = ( d/dx Series_Reversion( ( sinh(x)/cosh(x)^2 + atan(sinh(x)) )/2 ) )^(1/3).
%F A281181 E.g.f. C(x) and related series S(x) (e.g.f. of A281180) satisfy:
%F A281181 (1.a) C(x)^2 - S(x)^2 = 1.
%F A281181 (1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
%F A281181 Integrals.
%F A281181 (2.a) S(x) = Integral C(x)^4 dx.
%F A281181 (2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
%F A281181 Exponential.
%F A281181 (3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
%F A281181 (3.b) C(x) = cosh( Integral C(x)^3 dx ).
%F A281181 (3.c) S(x) = sinh( Integral C(x)^3 dx ).
%F A281181 Derivatives.
%F A281181 (4.a) S'(x) = C(x)^4.
%F A281181 (4.b) C'(x) = C(x)^3*S(x).
%F A281181 (4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
%F A281181 (4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).
%F A281181 Explicit Solutions.
%F A281181 (5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
%F A281181 (5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
%F A281181 (5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
%F A281181 (5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
%F A281181 (5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
%F A281181 (5.f) C(x)^4 = d/dx Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
%F A281181 (5.g) C(x)^5 = d/dx Series_Reversion( Integral C(i*x)^5 dx ).
%e A281181 E.g.f.: 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! + 39504564917358001*x^18/18! + 22726779729476308093*x^20/20! +...
%e A281181 such that
%e A281181 (1) C(x) = cosh( Integral C(x)^3 dx ),
%e A281181 (2) C(x)^2 - S(x)^2 = 1, and
%e A281181 (3) C(x) = 1 + Integral C(x)^3*S(x) dx,
%e A281181 where S(x) begins:
%e A281181 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! + 1000749157519458304*x^19/19! + 635146072839001735168*x^21/21! +...+ A281180(n)*x^(2*n-1)/(2*n-1)! +...
%e A281181 RELATED SERIES.
%e A281181 As power series with reduced fractional coefficients, S(x) and C(x) begin:
%e A281181 S(x) = x + 2/3*x^3 + 11/15*x^5 + 292/315*x^7 + 3548/2835*x^9 + 273766/155925*x^11 + 15360178/6081075*x^13 + 214706776/58046625*x^15 +...
%e A281181 C(x) = 1 + 1/2*x^2 + 13/24*x^4 + 493/720*x^6 + 37369/40320*x^8 + 4732249/3628800*x^10 + 901188997/479001600*x^12 + 240798388357/87178291200*x^14 +...
%e A281181 Related powers of series C(x) are given as follows.
%e A281181 C(x)^2 = 1 + 2*x^2/2! + 32*x^4/4! + 1376*x^6/6! + 114176*x^8/8! + 15519488*x^10/10! + 3132551168*x^12/12! + 879422726144*x^14/14! + 327670676455424*x^16/16! + 156439068819587072*x^18/18! +...+ A281183(n)*x^(2*n)/(2*n)! +...
%e A281181 where C(x)^2 = 1 + S(x)^2.
%e A281181 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! +...+ A281184(n)*x^(2*n)/(2*n)! +...
%e A281181 where C(x)^3 = d/dx log( C(x) + S(x) ).
%e A281181 Also, C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
%e A281181 C(x)^4 = 1 + 4*x^2/2! + 88*x^4/4! + 4672*x^6/6! + 454144*x^8/8! + 70084096*x^10/10! + 15728822272*x^12/12! + 4836914249728*x^14/14! + 1952137912385536*x^16/16! + 1000749157519458304*x^18/18! +...+ A281180(n+1)*x^(2*n)/(2*n)! +...
%e A281181 where C(x)^4 = d/dx S(x).
%t A281181 nMax = 30; m = maxExponent = 2*nMax; a[n_] := Module[{S = x, C = 1}, For[i = 1, i <= n, i++, S = Integrate[C^4 + x*O[x]^m // Normal, x] + O[x]^m // Normal; C = 1 + Integrate[S*C^3 + O[x]^m // Normal, x]] + O[x]^m // Normal; (2*n)!*Coefficient[C, x, 2*n]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, nMax}] (* _Jean-François Alcover_, Jan 20 2017, adapted from PARI *)
%t A281181 nmax = 20; Table[(CoefficientList[Sqrt[D[InverseSeries[Series[(2*x + Sin[2*x])/4, {x, 0, 2*nmax - 1}], x], x]], x] * Range[0, 2*nmax - 2]!)[[2*n - 1]], {n, 1, nmax}] (* _Vaclav Kotesovec_, Sep 02 2017 *)
%o A281181 (PARI) {a(n) = my(S=x,C=1); for(i=0,n, S = intformal( C^4 +x*O(x^(2*n))); C = 1 + intformal( S*C^3 ) ); (2*n)!*polcoeff(C,2*n)}
%o A281181 for(n=0,30,print1(a(n),", "))
%Y A281181 Cf. A281180 (S), A281182 (C+S), A281183 (C^2), A281184 (C^3), A001148, A100402, A122553.
%K A281181 nonn
%O A281181 0,3
%A A281181 _Paul D. Hanna_, Jan 16 2017
%E A281181 Name simplified by _Paul D. Hanna_, Jan 22 2017