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%I #10 May 18 2014 18:46:33
%S 1,1,1,4,9,25,64,256,729,2601,7921,28900,90000,318096,1016064,3888784,
%T 12694969,46881409,155825289,592435600,1987643889,7408217041,
%U 25153960000,95823678916,327811212304,1227814692624,4234507915264
%N a(n) is the square of the coefficient of x^n in 1/(1 - x*A(x^2)), where g.f. A(x) = Sum_{n>=0} a(n)*x^n.
%F a(n) = A121649(n)^2.
%F G.f.: A(x^2) = (1 - 1/B(x) )/x, where B(x) = Sum_{n>=0} a(n)^(1/2)*x^n is the g.f. of A121649.
%e A(x) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 25*x^5 + 64*x^6 + 256*x^7 +...
%e Take the square root of each term, a(n)^(1/2), and
%e let B(x) be the g.f. of the resulting sequence:
%e B(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 16*x^7 +...
%e Then 1/B(x) = 1 - x*A(x^2):
%e 1/B(x) = 1 - x - x^3 - x^5 - 4*x^7 - 9*x^9 - 25*x^11 - 64*x^13 -...
%o (PARI) {a(n)=if(n==0, 1, sum(k=0, n\2, polcoeff(x^(n-2*k)*(sum(j=0, k, a(j)*x^(2*j))+x*O(x^n))^(n-2*k), n))^2)}
%o (PARI) {a(n)=local(A,m); if(n<0, 0, m=1; A=1+x+O(x^2); while(m<=n, m*=2; A=1/(1-x*sum(k=0,m-1, polcoeff(A,k)^2*x^(2*k), O(x^(2*m))))); polcoeff(A,n)^2)} /* _Michael Somos_, Aug 18 2006 */
%Y Cf. A121649; bisection of A121649: A121650, A121651.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Aug 14 2006
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