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A266288 Expansion of a(q)^2 * (c(q)/3)^3 in powers of q where a(), c() are cubic AGM theta functions.

%I A266288 #16 Nov 09 2023 08:52:53
%S A266288 1,15,81,241,624,1215,2402,3855,6561,9360,14640,19521,28562,36030,
%T A266288 50544,61681,83520,98415,130322,150384,194562,219600,279840,312255,
%U A266288 390001,428430,531441,578882,707280,758160,923522,986895,1185840,1252800,1498848,1581201
%N A266288 Expansion of a(q)^2 * (c(q)/3)^3 in powers of q where a(), c() are cubic AGM theta functions.
%C A266288 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A266288 Convolution of A008653 and A106402.
%H A266288 G. C. Greubel, <a href="/A266288/b266288.txt">Table of n, a(n) for n = 1..5000</a>
%F A266288 a(n) is multiplicative with a(p^e) = ((p^4)^(e+1) - s^(e+1)) / (p^4 - s) where s = 0 if p = 3, s = 1 if p == 1 (mod 3), s = -1 if p == 2 (mod 3).
%F A266288 Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 4*Pi^5/(729*sqrt(3)) = 0.9694405... (A344778). - _Amiram Eldar_, Nov 09 2023
%e A266288 G.f. = x + 15*x^2 + 81*x^3 + 241*x^4 + 624*x^5 + 1215*x^6 + 2402*x^7 + ...
%t A266288 a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (With[{s = {1, -1, 0}[[Mod[#, 3, 1]]]}, ((#^4)^(#2 + 1) - s^(#2 + 1)) / (#^4 - s)] & @@@ FactorInteger[n])];
%o A266288 (PARI) {a(n) = my(A, U1, u3, U9); if( n<1, 0, n--; A = x * O(x^n); U1 = eta(x + A)^3; u3 = eta(x^3 + A); U9 = eta(x^9 + A)^3; polcoeff( U1 * u3^7 * (1 + 9*x*U9/U1)^2, n))};
%o A266288 (PARI) {a(n) = my(A, p, e, s); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, p^(4*e), s=-(-1)^(p%3);  ((p^4)^(e+1) - s^(e+1)) / (p^4 - s))))};
%o A266288 (Magma) A := Basis( ModularForms( Gamma1(3), 5),37);  A[2];
%Y A266288 Cf. A004016, A005882, A005928, A008653, A106402, A344778.
%K A266288 nonn,easy,mult
%O A266288 1,2
%A A266288 _Michael Somos_, Dec 26 2015