|
%I #18 Mar 12 2021 22:24:47
%S 1,-1,1,-2,2,-3,4,-6,7,-9,12,-14,18,-22,28,-34,41,-50,60,-72,86,-105,
%T 124,-146,174,-204,240,-282,332,-386,450,-524,606,-703,812,-940,1082,
%U -1243,1428,-1636,1873,-2140,2448,-2788,3172,-3610,4096,-4646,5264,-5962
%N Expansion of 1 / (chi(x) * chi(x^7)) in powers of x where chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A246762/b246762.txt">Table of n, a(n) for n = 0..1500</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^7) * eta(q^28) / (eta(q^2) * eta(q^14))^2 in powers of q.
%F Euler transform of period 28 sequence [ -1, 1, -1, 0, -1, 1, -2, 0, -1, 1, -1, 0, -1, 2, -1, 0, -1, 1, -1, 0, -2, 1, -1, 0, -1, 1, -1, 0, ...].
%F Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (v - u^2) - 2 * (u*v)^2 * (1 - u*v)^2.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = f(t) where q = exp(2 Pi i t).
%F G.f.: Product_{k>0} (1 + (-x)^k) * (1 + (-x)^(7*k)).
%F a(n) = (-1)^n * A093950(n).
%F Convolution inverse of A112212.
%e G.f. = 1 - x + x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 6*x^7 + 7*x^8 - 9*x^9 + ...
%e G.f. = q - q^4 + q^7 - 2*q^10 + 2*q^13 - 3*q^16 + 4*q^19 - 6*q^22 + 7*q^25 + ...
%t a[ n_] := SeriesCoefficient[ Product[ 1 + (-x)^k, {k, n}] Product[ 1 + (-x)^k, {k, 7, n, 7}], {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x] QPochhammer[ x^7, -x^7], {x, 0, n}];
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/3)* eta[q]*eta[q^4]*eta[q^7]*eta[q^28]/(eta[q^2]*eta[q^14])^2, {q,0,60}],q]; Table[a[[n]], {n,1,50}] (* _G. C. Greubel_, Jul 04 2018 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n)) * prod( k=1, n\7, 1 + (-x)^(7*k), 1 + x * O(x^n)), n))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A) / (eta(x^2 + A) * eta(x^14 + A))^2, n))};
%Y Cf. A093950, A112212.
%K sign
%O 0,4
%A _Michael Somos_, Sep 02 2014
|
