login
Expansion of (eta(q^3)^2 * eta(q^7) * eta(q^63)) / (eta(q) * eta(q^9) * eta(q^21)^2) in powers of q.
0

%I #17 Sep 08 2015 08:20:13

%S 1,1,2,1,3,3,4,3,5,6,9,8,11,12,15,16,18,21,26,28,35,39,47,51,58,66,77,

%T 85,97,108,125,141,156,174,195,218,245,270,304,336,377,417,467,512,

%U 573,627,702,770,853,935,1035,1136,1257,1371,1515,1654,1822,1989,2184,2382

%N Expansion of (eta(q^3)^2 * eta(q^7) * eta(q^63)) / (eta(q) * eta(q^9) * eta(q^21)^2) in powers of q.

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%F Expansion of ( (c(q) * b(q^3) * b(q^7) * c(q^21)) / (b(q) * c(q^3) * c(q^7) * b(q^21)) )^(1/4) in powers of q where b(), c() are cubic AGM theta functions.

%F Euler transform of a period 63 sequence.

%F G.f. A(x) = B(x) / B(x^7) where B() is the g.f. of A112194.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) = f(1/A(x), 1/A(x^2)) where f(u, v) = u^3 + v^3 - 2*u*v*(u + v) - u^2*v^2 - u*v.

%F G.f.: x * Product_{k>0} (1 - x^(3*k))^2 * (1 - x^(7*k)) * (1 - x^(63*k)) / ( (1 - x^k) * (1 - x^(9*k)) * (1 - x^(21*k))^2 ).

%F a(n) ~ exp(4*Pi*sqrt(n/7)/3) / (sqrt(6) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015

%e G.f. = q + q^2 + 2*q^3 + q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 3*q^8 + 5*q^9 + 6*q^10 + 9*q^11 + ...

%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3]^2 QPochhammer[ q^7] QPochhammer[ q^63] / (QPochhammer[ q] QPochhammer[ q^9] QPochhammer[ q^21]^2), {q, 0, n}]; (* _Michael Somos_, Apr 24 2015 *)

%t nmax = 60; Rest[CoefficientList[Series[x * Product[(1 - x^(3*k))^2 * (1 - x^(7*k)) * (1 - x^(63*k)) / ( (1 - x^k) * (1 - x^(9*k)) * (1 - x^(21*k))^2 ), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 08 2015 *)

%o (PARI) {a(n) = my(A, u, v); if( n<0, 0, A = x; for( k=2, n, u = A + x * O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff( u^3 + v^3 - 2*u*v*(u + v) - u^2*v^2 - u*v, k+2) / 2); polcoeff(A, n))};

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^7 + A) * eta(x^63 + A) / (eta(x + A) * eta(x^9 + A) * eta(x^21 + A)^2), n))};

%Y Cf. A112194.

%K nonn

%O 1,3

%A _Michael Somos_, Mar 17 2004