A261866 - OEIS

%I #11 Mar 12 2021 22:24:48

%S 1,0,2,2,5,6,13,16,28,38,60,80,122,162,234,312,436,576,789,1032,1386,

%T 1802,2381,3070,4008,5128,6618,8414,10752,13576,17210,21592,27162,

%U 33890,42340,52538,65244,80544,99458,122208,150126,183634,224527,273480,332898

%N Expansion of (G(-x) / chi(-x))^2 in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function.

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

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

%H G. C. Greubel, <a href="/A261866/b261866.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 Euler transform of period 20 sequence [ 0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 0, ...].

%F Expansion of (f(x^4, x^6) / f(-x^2, -x^3))^2 in powers of x where f(, ) is Ramanujan's general theta function.

%F Expansion of (f(-x, -x^9) * f(-x^8, -x^12) / (f(-x) * f(-x^20)))^2 in powers of x where f(, ) is Ramanujan's general theta function.

%F Expansion of (f(-x^2, x^3) / phi(-x^2))^2 in powers of x where phi() is a Ramanujan theta function.

%F G.f.: (Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k))))^2.

%F a(n) = A147699(5*n).

%F Convolution square of A122134.

%e G.f. = 1 + 2*x^2 + 2*x^3 + 5*x^4 + 6*x^5 + 13*x^6 + 16*x^7 + 28*x^8 + ...

%e G.f. = q + 2*q^41 + 2*q^61 + 5*q^81 + 6*q^101 + 13*q^121 + 16*q^141 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4, -x^5] QPochhammer[ -x, -x^5] QPochhammer[ x, x^2])^-2, {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 0}[[Mod[k, 20, 1]]], {k, n}], {x, 0, n}];

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0][k%20 + 1]), n))};

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n + 1) - 1)\2, x^(k^2 + k) / prod(i=1, 2*k, 1 - x^i, 1 + x * O(x^(n - k^2-k))))^2, n))};

%Y Cf. A122134, A147699.

%K nonn

%O 0,3

%A _Michael Somos_, Sep 03 2015