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A299855 G.f. C(x)^(1/2) satisfies C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)*S(x)^(1/2) = S'(x)*C(x)^(1/2) = 72*x.

%I A299855 #11 Apr 05 2019 12:00:06
%S A299855 1,6,-12,60,-384,2772,-21504,175032,-1474560,12748164,-112459776,
%T A299855 1008263880,-9160359936,84151254600,-780341870592,7294711613040,
%U A299855 -68670084612096,650409360439140,-6193772337561600,59267126633699880,-569566264641454080,5494909312181603160,-53198968510406983680,516695227418183158800,-5033085020810678108160
%N A299855 G.f. C(x)^(1/2) satisfies C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)*S(x)^(1/2) = S'(x)*C(x)^(1/2) = 72*x.
%C A299855 a(n) = -(-1)^n * A244038(n) / (3*n-2) for n>=1.
%F A299855 The functions C = C(x) and S = S(x) satisfy:
%F A299855 (1a) sqrt(C) - sqrt(S) = 1.
%F A299855 (1b) C'*sqrt(S) = S'*sqrt(C) = 72*x.
%F A299855 (1c) C' = 72*x/sqrt(S).
%F A299855 (1d) S' = 72*x/sqrt(C).
%F A299855 Integrals.
%F A299855 (2a) C = 1 + Integral 72*x/sqrt(S) dx.
%F A299855 (2b) S = Integral 72*x/sqrt(C) dx.
%F A299855 (2c) C = 1 + Integral S'*sqrt(C/S) dx.
%F A299855 (2d) S = Integral C'*sqrt(S/C) dx.
%F A299855 Exponentials.
%F A299855 (3a) sqrt(C) = exp( Integral 36*x/(C*sqrt(S)) dx ).
%F A299855 (3b) sqrt(S) = 6*x*exp( Integral 36*x/(S*sqrt(C)) - 1/x dx ).
%F A299855 (3c) C - S = exp( Integral 72*x/(C*sqrt(S) + S*sqrt(C)) dx ).
%F A299855 (3d) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).
%F A299855 Functional equations.
%F A299855 (4a) C = 1/3 - 36*x^2 + (2/3)*C^(3/2).
%F A299855 (4b) S = 36*x^2 - (2/3)*S^(3/2).
%F A299855 Explicit solutions.
%F A299855 (5a) C(x) = 1 + Sum_{n>=1} 2*(-4)^n*binomial(3*n/2,n)/((3*n-2)*(3*n-4)) * x^n.
%F A299855 (5b) S(x) = 36*x^2 + Sum_{n>=3} 18*(-4)^n*(3*n-3)*binomial(3*n/2-2,n)/((3*n-4)*(3*n-6)) * x^n.
%F A299855 (5c) sqrt(C(x)) = 1 + Sum_{n>=1} -(-4)^n * binomial(3*n/2,n)/(3*n-2) * x^n.
%F A299855 Formulas for terms.
%F A299855 a(n) = -(-4)^n * binomial(3*n/2,n) / (3*n-2) for n>=1, with a(0) = 1.
%e A299855 G.f.: C(x)^(1/2) = 1 + 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
%e A299855 RELATED SERIES.
%e A299855 C(x) = 1 + 12*x + 12*x^2 - 24*x^3 + 96*x^4 - 504*x^5 + 3072*x^6 - 20592*x^7 + 147456*x^8 - 1108536*x^9 + 8650752*x^10 + ...
%e A299855 S(x) = 36*x^2 - 144*x^3 + 864*x^4 - 6048*x^5 + 46080*x^6 - 370656*x^7 + 3096576*x^8 - 26604864*x^9 + 233570304*x^10 + ...
%e A299855 sqrt(S(x)) = 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
%e A299855 where C(x)^(1/2) - S(x)^(1/2) = 1
%e A299855 and C'*sqrt(S) = S'*sqrt(C) = 72*x.
%o A299855 (PARI) {a(n) = my(C=1, S=x^2); for(i=0, n, C = 1 + intformal( 72*x/sqrt(S +x^3*O(x^n)) ); S = intformal( 72*x/sqrt(C) ) ); polcoeff(sqrt(C), n)}
%o A299855 for(n=0,30,print1(a(n),", "))
%o A299855 (PARI) {a(n) = if(n==0,1, -(-4)^n * binomial(3*n/2,n) / (3*n-2) )}
%o A299855 for(n=0,30,print1(a(n),", "))
%Y A299855 Cf. A299853, A299854, A244038.
%K A299855 sign
%O A299855 0,2
%A A299855 _Paul D. Hanna_, Feb 20 2018