This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A281182 #32 Sep 03 2017 04:42:04
%S A281182 1,1,1,4,13,88,493,4672,37369,454144,4732249,70084096,901188997,
%T A281182 15728822272,240798388357,4836914249728,85948640603761,
%U A281182 1952137912385536,39504564917358001,1000749157519458304,22726779729476308093,635146072839001735168,15998009117983994065693,488855521088102855606272,13526765851190230940840809,448599416591747486039670784,13528070218935445806530640649
%N 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.
%H A281182 Paul D. Hanna, <a href="/A281182/b281182.txt">Table of n, a(n) for n = 0..200</a>
%F A281182 E.g.f. exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
%F A281182 E.g.f. exp( Series_Reversion( ( sinh(x)/cosh(x)^2 + atan(sinh(x)) )/2 ) ).
%F A281182 E.g.f. C(x) + S(x) where related series S(x) and C(x) satisfy:
%F A281182 (1.a) C(x)^2 - S(x)^2 = 1.
%F A281182 (1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
%F A281182 Integrals.
%F A281182 (2.a) S(x) = Integral C(x)^4 dx.
%F A281182 (2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
%F A281182 Exponential.
%F A281182 (3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
%F A281182 (3.b) C(x) = cosh( Integral C(x)^3 dx ).
%F A281182 (3.c) S(x) = sinh( Integral C(x)^3 dx ).
%F A281182 Derivatives.
%F A281182 (4.a) S'(x) = C(x)^4.
%F A281182 (4.b) C'(x) = C(x)^3*S(x).
%F A281182 (4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
%F A281182 (4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).
%F A281182 Explicit Solutions.
%F A281182 (5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
%F A281182 (5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
%F A281182 (5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
%F A281182 (5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
%F A281182 (5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
%e A281182 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! +...
%e A281182 where A(x) = C(x) + S(x) and the series for C(x) and S(x) begin:
%e A281182 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)! +...
%e A281182 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)! +...
%e A281182 such that C(s) + S(x) = exp( Integral C(x)^3 dx ).
%e A281182 The logarithm of the e.g.f. begins:
%e A281182 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! +...
%e A281182 which equals Integral C(x)^3 dx.
%e A281182 Also, log(C(x) + S(x)) = Series_Reversion( Integral 1/cosh(x)^3 dx ).
%t A281182 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 *)
%o A281182 (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)}
%o A281182 for(n=0, 30, print1(a(n), ", "))
%o A281182 (PARI) /* From S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ) */
%o A281182 {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)}
%o A281182 for(n=0, 30, print1(a(n), ", "))
%Y A281182 Cf. A281180 (S), A281181 (C), A281183 (C^2), A281184 (C^3).
%K A281182 nonn
%O A281182 0,4
%A A281182 _Paul D. Hanna_, Jan 16 2017