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A172390
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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.
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2
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1, 8, 24, 0, -168, 0, 2112, 0, -32040, 0, 536256, 0, -9542976, 0, 177126912, 0, -3390361128, 0, 66436117440, 0, -1326185205696, 0, 26872637815296, 0, -551301904867392, 0, 11428295231789568, 0, -239010764560888320, 0
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OFFSET
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0,2
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LINKS
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FORMULA
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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.
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) ).
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]
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EXAMPLE
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G.f.: A(x) = 1 + 8*x + 24*x^2 - 168*x^4 + 2112*x^6 - 32040*x^8 + ...
A(x) = G(x/A(x))^2 where G(x) = 1/AGM(1, (1-16x)^(1/2)) is the power series:
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 + ...
The square root of g.f. A(x) begins:
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 + ...
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PROG
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(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)}
(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
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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