A226635 - OEIS

%I #20 Oct 04 2023 16:05:53

%S 1,1,2,3,6,8,13,18,27,37,53,71,100,132,179,235,313,405,531,681,880,

%T 1119,1429,1801,2280,2852,3575,4444,5529,6827,8436,10357,12716,15530,

%U 18958,23036,27978,33839,40896,49254,59265,71083,85180,101781,121494,144659

%N Expansion of psi(x^4) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A226635/b226635.txt">Table of n, a(n) for n = 0..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^(-11/24) * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q.

%F Euler transform of period 8 sequence [1, 1, 1, 2, 1, 1, 1, 0, ...].

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

%F 2 * a(n) = A073252(2*n + 1).  -2 * a(n) = A022597(2*n + 1).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(13/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017

%F Expansion of (chi(q)^2 - chi(-q)^2)/(4*q) in powers of q^2 where chi() is a Ramanujan theta function. - _Michael Somos_, Nov 02 2019

%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 13*x^6 + 18*x^7 + 27*x^8 + 37*x^9 + ...

%e G.f. = q^11 + q^35 + 2*q^59 + 3*q^83 + 6*q^107 + 8*q^131 + 13*q^155 + 18*q^179 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^2] / (2 q^(1/2) QPochhammer[ q]), {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A)), n))};

%Y Cf. A022597, A073252.

%K nonn

%O 0,3

%A _Michael Somos_, Aug 31 2013