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G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(3*n+1).
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%I #9 Feb 02 2025 14:16:24

%S 1,1,4,23,151,1074,8059,62814,503619,4126954,34411602,291025337,

%T 2490377810,21523367553,187603609077,1647252368595,14556722879278,

%U 129366008725176,1155458240271571,10366549508487178,93382085749705066,844255894224907354

%N G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(3*n+1).

%F G.f. A(x) satisfies the continued fraction:

%F 1 = A(x)/(1+ x*A(x)^3/(1- x*(1+x)*A(x)^3/(1+ x^3*A(x)^3/(1+ x^2*(1-x^2)*A(x)^3/(1+ x^5*A(x)^3/(1- x^3*(1+x^3)*A(x)^3/(1+ x^7*A(x)^3/(1+ x^4*(1-x^4)*A(x)^3/(1- ...)))))))))

%F due to an identity of a partial elliptic theta function.

%e G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 151*x^4 + 1074*x^5 + 8059*x^6 +...

%e which satisfies:

%e 1 = A(x) - x*A(x)^4 - x^3*A(x)^7 + x^6*A(x)^10 + x^10*A(x)^13 - x^15*A(x)^16 - x^21*A(x)^19 ++--...

%e Related expansions.

%e A(x)^4 = 1 + 4*x + 22*x^2 + 144*x^3 + 1025*x^4 + 7696*x^5 +...

%e A(x)^7 = 1 + 7*x + 49*x^2 + 364*x^3 + 2814*x^4 + 22400*x^5 +...

%o (PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(2*(#A))+1, (-x)^(m*(m+1)/2)*Ser(A)^(3*m+1)), #A-1)); if(n<0, 0, A[n+1])}

%Y Cf. A193111, A193112, A193114, A193115, A193116.

%K nonn,changed

%O 0,3

%A _Paul D. Hanna_, Jul 16 2011