A259357 - OEIS

%I #15 Mar 12 2021 22:24:48

%S 1,1,1,1,2,1,2,2,3,3,3,3,5,5,5,6,7,7,9,9,11,11,13,13,16,17,19,20,23,

%T 24,27,29,32,34,38,40,46,48,52,56,62,65,72,76,84,89,97,102,113,119,

%U 129,137,149,157,171,181,196,208,224,236,256,270,290,308,331

%N Expansion of f(-x^5)^2 / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.

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

%C Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

%D G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, 2012, see p. 12, Entry 2.1.3, Equation (2.1.21).

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 3.

%H G. C. Greubel, <a href="/A259357/b259357.txt">Table of n, a(n) for n = 0..2500</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 f(-x^5) * f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function.

%F Expansion of f(-x^5) * G(x) in powers of x where f() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - _Michael Somos_, Jul 09 2015

%F Euler transform of period 5 sequence [ 1, 0, 0, 1, -1, ...].

%F G.f.: Product_{k>0} (1 - x^(5*k)) / ((1 - x^(5*k-4)) * (1 - x^(5*k-1))).

%F Convolution of A035959 and A113428.

%e G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + ...

%e G.f. = q^23 + q^143 + q^263 + q^383 + 2*q^503 + q^623 + 2*q^743 + 2*q^863 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}];

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -1, 0, 0, -1][k%5+1]), n))};

%Y Cf. A035959, A113428, A259358.

%K nonn

%O 0,5

%A _Michael Somos_, Jun 24 2015