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A172390 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(x*G(x)^2) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n. 2

%I #5 Feb 02 2018 02:49:54

%S 1,8,24,0,-168,0,2112,0,-32040,0,536256,0,-9542976,0,177126912,0,

%T -3390361128,0,66436117440,0,-1326185205696,0,26872637815296,0,

%U -551301904867392,0,11428295231789568,0,-239010764560888320,0

%N G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(x*G(x)^2) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n.

%F G.f.: A(x) = x/Series_Reversion(x*G(x)^2)) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n = 1/agm(1, (1-16*x)^(1/2)) = g.f. of A002894 and G(x)^2 is the g.f. of A036917.

%F Self-convolution of A158101, which is a bisection of A158100; A158100 has g.f. F(x) that satisfies: F(x) = 1/AGM(1, 1 - 8*x/F(x) ).

%F a(n) = [x^n] AGM(1,(1-16x)^(1/2))^(2n-2)/(1-n) for n>1 where AGM is the arithmetic-geometric mean of Gauss. - _Paul D. Hanna_, Mar 20 2010]

%e G.f.: A(x) = 1 + 8*x + 24*x^2 - 168*x^4 + 2112*x^6 - 32040*x^8 + ...

%e A(x) = G(x/A(x))^2 where G(x) = 1/AGM(1, (1-16x)^(1/2)) is the power series:

%e G(x) = 1 + 2^2*x + 6^2*x^2 + 20^2*x^3 + 70^2*x^4 + 252^2*x^5 + ... + C(2n,n)^2*x^n + ...

%e The square root of g.f. A(x) begins:

%e A(x)^(1/2) = 1 + 4*x + 4*x^2 - 16*x^3 - 28*x^4 + 176*x^5 + 336*x^6 + ... + A158101(n)*x^n + ...

%o (PARI) {a(n)=local(G=sum(m=0,n,binomial(2*m,m)^2*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G^2),n)}

%o (PARI) {a(n)=if(n==1,8,polcoeff(agm(1,sqrt(1-16*x +x^2*O(x^n)))^(2*n-2),n)/(1-n))} \\ _Paul D. Hanna_, Mar 20 2010

%Y Cf. A036917, A002894, A158101, A158100.

%K sign

%O 0,2

%A _Paul D. Hanna_, Feb 04 2010

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