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A281180 E.g.f. S(x) satisfies: S(x) = Integral (1 + S(x)^2)^2 dx.

%I A281180 #43 Mar 08 2025 06:02:52
%S A281180 1,4,88,4672,454144,70084096,15728822272,4836914249728,
%T A281180 1952137912385536,1000749157519458304,635146072839001735168,
%U A281180 488855521088102855606272,448599416591747486039670784,483861305506660094099058589696,606050665000453965359938841608192,872366179652871528356910686198038528,1430068361869553198039835379199635357696,2648687881942689612933392158083076801429504,5503854158077547090902251582359116752300802048
%N A281180 E.g.f. S(x) satisfies: S(x) = Integral (1 + S(x)^2)^2 dx.
%H A281180 Paul D. Hanna, <a href="/A281180/b281180.txt">Table of n, a(n) for n = 1..100</a>
%F A281180 E.g.f. S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
%F A281180 E.g.f. S(x) = Series_Reversion( ( x/(1+x^2) + atan(x) )/2 ).
%F A281180 E.g.f. S(x) and related series C(x) (e.g.f. of A281181) satisfy:
%F A281180 (1.a) C(x)^2 - S(x)^2 = 1.
%F A281180 (1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
%F A281180 Integrals.
%F A281180 (2.a) S(x) = Integral C(x)^4 dx.
%F A281180 (2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
%F A281180 Exponential.
%F A281180 (3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
%F A281180 (3.b) C(x) = cosh( Integral C(x)^3 dx ).
%F A281180 (3.c) S(x) = sinh( Integral C(x)^3 dx ).
%F A281180 Derivatives.
%F A281180 (4.a) S'(x) = C(x)^4.
%F A281180 (4.b) C'(x) = C(x)^3*S(x).
%F A281180 (4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
%F A281180 (4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).
%F A281180 Explicit Solutions.
%F A281180 (5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
%F A281180 (5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
%F A281180 (5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
%F A281180 (5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
%F A281180 (5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
%e A281180 E.g.f.: 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! +...
%e A281180 such that
%e A281180 (1) C(x)^2 - S(x)^2 = 1, and
%e A281180 (2) S'(x) = C(x)^4,
%e A281180 where C(x) begins:
%e A281180 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! +...+ A281181(n)*x^(2*n)/(2*n)! +...
%e A281180 RELATED SERIES.
%e A281180 As power series with reduced fractional coefficients, S(x) and C(x) begin:
%e A281180 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 A281180 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 A281180 The series reversion of the e.g.f. begins:
%e A281180 Series_Reversion(S(x)) = x - 2/3*x^3 + 3/5*x^5 - 4/7*x^7 + 5/9*x^9 - 6/11*x^11 + 7/13*x^13 - 8/15*x^15 +...
%e A281180 which equals ( x/(1+x^2) + atan(x) )/2.
%e A281180 Related powers of series C(x) are given as follows.
%e A281180 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 A281180 where C(x)^2 = 1 + S(x)^2.
%e A281180 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 A281180 where C(x)^3 = d/dx log( C(x) + S(x) ).
%e A281180 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! +...
%e A281180 where C(x)^4 = d/dx S(x).
%t A281180 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 - 1)!*Coefficient[S, x, 2*n - 1]]; Table[an = a[n]; Print[ "a(", n, ") = ", an]; an, {n, 1, nMax}] (* _Jean-François Alcover_, Jan 20 2017, adapted from first PARI program *)
%t A281180 nmax = 20; Table[(CoefficientList[InverseSeries[Series[(x/(1 + x^2) + ArcTan[x])/2, {x, 0, 2*nmax - 1}], x], x] * Range[0, 2*nmax - 1]!)[[2*n]], {n, 1, nmax}] (* _Vaclav Kotesovec_, Sep 02 2017 *)
%o A281180 (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-1)!*polcoeff(S,2*n-1)}
%o A281180 for(n=1,30,print1(a(n),", "))
%o A281180 (PARI) /* S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ) */
%o A281180 {a(n) = my(S=x); S = serreverse( intformal( 1/(1 + x^2 +x*O(x^(2*n)))^2)); (2*n-1)!*polcoeff(S,2*n-1)}
%o A281180 for(n=1,30,print1(a(n),", "))
%Y A281180 Cf. A281181 (C), A281182 (C+S), A281183 (C^2), A281184 (C^3).
%K A281180 nonn
%O A281180 1,2
%A A281180 _Paul D. Hanna_, Jan 16 2017
%E A281180 Name simplified by _Paul D. Hanna_, Jan 22 2017