A226635 Expansion of psi(x^4) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

1, 1, 2, 3, 6, 8, 13, 18, 27, 37, 53, 71, 100, 132, 179, 235, 313, 405, 531, 681, 880, 1119, 1429, 1801, 2280, 2852, 3575, 4444, 5529, 6827, 8436, 10357, 12716, 15530, 18958, 23036, 27978, 33839, 40896, 49254, 59265, 71083, 85180, 101781, 121494, 144659

Offset

0,3

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-11/24) * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q.

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

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

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

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(13/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

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

Example

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

Mathematica

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

Prog

(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))};

Crossrefs

Cf. A022597, A073252.

Sequence in context: A240076 A266771 A295342 * A024788 A285472 A318027

Adjacent sequences: A226632 A226633 A226634 * A226636 A226637 A226638

Keyword

nonn

Author

Michael Somos, Aug 31 2013

Status

approved