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%I #27 Jan 22 2017 13:56:45
%S 1,2,32,1376,114176,15519488,3132551168,879422726144,327670676455424,
%T 156439068819587072,93116847688811282432,67602541384815095054336,
%U 58796336342280763841970176,60351125684887424790500999168,72187248798124538021926003539968,99529442030183464236437157900713984,156697512616609083360755035696397287424
%N E.g.f. C(x)^2 = cosh( Integral C(x)^3 dx )^2 where C(x) is described by A281181.
%H Paul D. Hanna, <a href="/A281183/b281183.txt">Table of n, a(n) for n = 0..100</a>
%F E.g.f. C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
%F E.g.f. C(x)^2 = d/dx Series_Reversion( (2*x + sin(2*x))/4 ).
%F E.g.f. C(x)^2 where related series S(x) and C(x) satisfy:
%F (1.a) C(x)^2 - S(x)^2 = 1.
%F (1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
%F Integrals.
%F (2.a) S(x) = Integral C(x)^4 dx.
%F (2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
%F Exponential.
%F (3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
%F (3.b) C(x) = cosh( Integral C(x)^3 dx ).
%F (3.c) S(x) = sinh( Integral C(x)^3 dx ).
%F Derivatives.
%F (4.a) S'(x) = C(x)^4.
%F (4.b) C'(x) = C(x)^3*S(x).
%F (4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
%F (4.d) (C(x)^4 - S(x)^4)' = 4*C(x)^4*S(x).
%F Explicit Solutions.
%F (5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
%F (5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
%F (5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
%F (5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
%F (5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
%e E.g.f.: 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! +...
%e such that C(x)^2 = 1 + S(x)^2 and the series for C(x) and S(x) begin:
%e 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 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)! +...
%o (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)!*polcoeff(C^2, 2*n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* From C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ) ) */
%o {a(n) = my(C2=x); C2 = deriv( serreverse( intformal( cos(x +x*O(x^(2*n)))^2 ))); (2*n)!*polcoeff(C2, 2*n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A281180 (S), A281181 (C), A281182 (C+S), A281184 (C^3).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 16 2017
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