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%I #7 Oct 14 2024 10:56:20
%S 1,1,4,20,141,1045,8172,66681,560173,4815143,42154010,374555139,
%T 3369272574,30623456589,280807674622,2594626014807,24133855340952,
%U 225796163598383,2123515433280699,20063236095299562,190347878004477651,1812680623454591643,17320874611556481567,166020895158801131827
%N G.f. A(x) satisfies (3*A(x^9)/A(x) - 1)^3 = 9*A(x^3)^3/A(x)^3 - 1 - 9*x.
%C Compare to the following cubic modular identity of Jacobi theta functions,
%C (3*H(q^9)/H(q) - 1)^3 = 9*H(q^3)^4/H(q)^4 - 1,
%C which holds for H(q) = theta_2(q), H(q) = theta_3(q), and H(q) = theta_4(q) (see Mathworld link).
%H Paul D. Hanna, <a href="/A376528/b376528.txt">Table of n, a(n) for n = 0..730</a>
%H Weisstein, Eric W., <a href="https://mathworld.wolfram.com/ModularEquation.html">Modular Equation.</a> From MathWorld -- A Wolfram Web Resource.
%e G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 141*x^4 + 1045*x^5 + 8172*x^6 + 66681*x^7 + 560173*x^8 + 4815143*x^9 + 42154010*x^10 + ...
%e RELATED SERIES.
%e (3*A(x^9)/A(x) - 1)^3 = 8 - 36*x - 54*x^2 - 171*x^3 - 1809*x^4 - 12258*x^5 - 92529*x^6 - 751626*x^7 + ...
%e 9*A(x^3)^3 / A(x)^3 = 9 - 27*x - 54*x^2 - 171*x^3 - 1809*x^4 - 12258*x^5 - 92529*x^6 - 751626*x^7 + ...
%e so that 9*A(x^3)^3 / A(x)^3 - (3*A(x^9)/A(x) - 1)^3 = 1 + 9*x.
%o (PARI) {a(n) = my(V=[1], A); for(m=0,n, V = concat(V,0); A = Ser(V);
%o V[#V] = polcoef( 1 + 9*x + (3*subst(A,x,x^9)/A - 1)^3 - 9*subst(A,x,x^3)^3/A^3, #V-1)/9 ); V[n+1]}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A376525, A376526.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 14 2024