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A289695 E.g.f.: S(x) satisfies: S(x) = Integral sqrt( (1 + S(x)^2)*(1 + 2*S(x)^2) ) dx.

%I A289695 #24 Mar 14 2025 02:35:56
%S A289695 1,3,33,819,34209,2189187,198717057,24278289651,3842052205761,
%T A289695 764478393601923,186805627856569953,54994092004290217779,
%U A289695 19197418751181422089569,7840711973025043515377667,3704137338316764145483007937,2004220869541285849551954747891
%N A289695 E.g.f.: S(x) satisfies: S(x) = Integral sqrt( (1 + S(x)^2)*(1 + 2*S(x)^2) ) dx.
%H A289695 Vaclav Kotesovec, <a href="/A289695/b289695.txt">Table of n, a(n) for n = 1..228</a>
%F A289695 E.g.f.: S(x) = Series_Reversion( Integral 1/sqrt( (1 + x^2)*(1 + 2*x^2) ) dx ).
%F A289695 E.g.f.: S(x) = sinh( Series_Reversion( Integral 1/sqrt( cosh(2*x) ) dx ) ).
%F A289695 E.g.f.: S(x) = Integral sqrt(1 + S(i*x)^2) / (1 + 2*S(i*x)^2) dx, where i^2 = -1.
%F A289695 Let C(x) be the e.g.f. of A193543, then
%F A289695 (1) S(x) = sqrt(C(x)^2 - 1),
%F A289695 (2) S(x) = Integral C(x) * sqrt(C(x)^2 + S(x)^2) dx,
%F A289695 (3) C(x) = 1 + Integral S(x) * sqrt(C(x)^2 + S(x)^2) dx,
%F A289695 (4) C(x) + S(x) = exp( Integral sqrt(1 + 2*S(x)^2) dx ).
%e A289695 E.g.f.: S(x) = x + 3*x^3/3! + 33*x^5/5! + 819*x^7/7! + 34209*x^9/9! + 2189187*x^11/11! + 198717057*x^13/13! + 24278289651*x^15/15! + 3842052205761*x^17/17! + 764478393601923*x^19/19! + 186805627856569953*x^21/21! +...
%e A289695 where
%e A289695 sqrt((1 + S(x)^2)*(1 + 2*S(x)^2)) = 1 + 3*x^2/2! + 33*x^4/4! + 819*x^6/6! + 34209*x^8/8! +...
%e A289695 RELATED SERIES.
%e A289695 C(x) = sqrt(1 + S(x)^2) equals the e.g.f. of A193543, and begins
%e A289695 C(x) = 1 + x^2/2! + 9*x^4/4! + 153*x^6/6! + 4977*x^8/8! + 261009*x^10/10! + 20039481*x^12/12! + 2121958377*x^14/14! + 296297348193*x^16/16! +...
%t A289695 a[ n_] := If[n<1, 0, (2*n-1)!*SeriesCoefficient[JacobiSN[I*x, 2]/I, {x, 0, 2*n-1}]]; (* _Michael Somos_, Feb 13 2025 *)
%o A289695 (PARI) {a(n) = my(S=1); S = sinh( serreverse( intformal( 1/sqrt( cosh(2*x +O(x^(2*n+1))) ) ) ) ); (2*n-1)!*polcoeff(S,2*n-1)}
%o A289695 for(n=1,20,print1(a(n),", "))
%Y A289695 Cf. A193543.
%K A289695 nonn
%O A289695 1,2
%A A289695 _Paul D. Hanna_, Aug 14 2017