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A226635 Expansion of psi(x^4) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions. 2
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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

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Last modified October 1 08:39 EDT 2020. Contains 337442 sequences. (Running on oeis4.)