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G.f.: A(x) = ( 1 + Sum_{n>=1} 2*(-2)^n * x^(n^2) )^(-1/2).
1

%I #8 Mar 30 2012 18:37:28

%S 1,2,6,20,66,228,804,2872,10374,37780,138436,509832,1885396,6996472,

%T 26039592,97160592,363332534,1361320284,5109306148,19205627608,

%U 72292274076,272454449560,1027979651384,3882579544656,14677889406396,55536827825096

%N G.f.: A(x) = ( 1 + Sum_{n>=1} 2*(-2)^n * x^(n^2) )^(-1/2).

%C Compare the g.f. to the theta_3 series: 1 + Sum_{n>=1} 2*x^(n^2).

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 66*x^4 + 228*x^5 + 804*x^6 +...

%e where

%e 1/A(x)^2 = 1 - 4*x + 8*x^4 - 16*x^9 + 32*x^16 - 64*x^25 + 128*x^36 - 256*x^49 + 512*x^64 +...+ 2*(-2)^n*x^(n^2) +...

%o (PARI) {a(n)=local(S=1+sum(m=1,sqrtint(n),2*(-2)^m*x^(m^2))+x*O(x^n));polcoeff(S^(-1/2),n)}

%Y Cf. A193235.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 18 2011