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%I #9 Oct 14 2024 14:45:47
%S 1,1,7,101,1182,16303,243618,3794541,61004863,1004801055,16865731633,
%T 287434810434,4960389320078,86507430659736,1522194102412522,
%U 26991460147758051,481824329994858705,8651729550012408710,156162244332280459203,2831802125153075595781,51565152204237125945253
%N G.f. A(x) satisfies (4*A(x^9)/A(x) - 1)^3 = 28*A(x^3)^4/A(x)^4 - 1 + 4*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="/A376526/b376526.txt">Table of n, a(n) for n = 0..500</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 + 7*x^2 + 101*x^3 + 1182*x^4 + 16303*x^5 + 243618*x^6 + 3794541*x^7 + 61004863*x^8 + 1004801055*x^9 + 16865731633*x^10 + ...
%e RELATED SERIES.
%e (4*A(x^9)/A(x) - 1)^3 = 27 - 108*x - 504*x^2 - 7840*x^3 - 73332*x^4 - 996240*x^5 - 14618520*x^6 + ...
%e 28*A(x^3)^4 / A(x)^4 = 28 - 112*x - 504*x^2 - 7840*x^3 - 73332*x^4 - 996240*x^5 - 14618520*x^6 + ...
%e so that 28*A(x^3)^4/A(x)^4 - (4*A(x^9)/A(x) - 1)^3 = 1 - 4*x.
%o (PARI) {a(n) = my(V=[1], A); for(m=0,n, V = concat(V,0); A = Ser(V);
%o V[#V] = polcoef( 28*subst(A,x,x^3)^4/A^4 - 1 + 4*x - (4*subst(A,x,x^9)/A - 1)^3, #V-1)/4 ); V[n+1]}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A376525.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 14 2024