login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Expansion of eta(q^4) * eta(q^28) / (eta(q) * eta(q^7)) in powers of q.
3

%I #19 Mar 12 2021 22:24:44

%S 1,1,2,3,4,6,9,13,17,24,32,42,56,73,96,123,158,201,254,320,402,504,

%T 624,774,955,1172,1436,1755,2138,2592,3140,3789,4560,5478,6564,7851,

%U 9362,11146,13240,15696,18574,21942,25880,30456,35796,42000,49196,57546

%N Expansion of eta(q^4) * eta(q^28) / (eta(q) * eta(q^7)) in powers of q.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A123648/b123648.txt">Table of n, a(n) for n = 1..1000</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 / (chi(-q) * chi(-q^2) * chi(-q^7) * chi(-q^14)) in powers of q where chi() is a Ramanujan theta function. - _Michael Somos_, Nov 11 2013

%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 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v*(1 + 2*(u+v) + 4*u*v).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u*v * (1 + 2*u) * (1 + 2*v) * (1 + u+v + 4*u*v). - _Michael Somos_, Nov 11 2013

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A161970. - _Michael Somos_, Nov 11 2013

%F 2 * a(n) = A123862(n) unless n=0. a(2*n) = A120006(n). - _Michael Somos_, Nov 11 2013

%F Convolution inverse of A161970. - _Michael Somos_, Nov 11 2013

%F a(n) ~ exp(2*Pi*sqrt(n/7)) / (8 * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 03 2018

%e G.f. = q + q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 6*q^6 + 9*q^7 + 13*q^8 + 17*q^9 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^4] QPochhammer[ q^28] / (QPochhammer[ q] QPochhammer[ q^7]), {q, 0, n}]

%t a[ n_] := SeriesCoefficient[ QPochhammer[-q, q] QPochhammer[-q^2, q^2] QPochhammer[-q^7, q^7] QPochhammer[-q^14, q^14], {q, 0, n}]

%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^28 + A) / (eta(x + A) * eta(x^7 + A)), n))}

%Y Cf. A120006, A123862, A161970.

%K nonn

%O 1,3

%A _Michael Somos_, Oct 04 2006