|
| |
|
|
A085140
|
|
Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q.
|
|
1
| |
|
|
1, 2, 2, 4, 5, 6, 10, 12, 15, 20, 26, 32, 40, 50, 60, 76, 92, 110, 134, 160, 191, 230, 272, 320, 380, 446, 522, 612, 715, 830, 966, 1120, 1292, 1494, 1720, 1976, 2272, 2602, 2974, 3400, 3876, 4412, 5020, 5700, 6460, 7322, 8282, 9352, 10559, 11900, 13396
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
In the notation of Dragonette the generating function is G_2(q^(1/2))/2.
Equals A000009 convolved with A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 16 2010]
|
|
|
REFERENCES
| L. A. Dragonette, Some asymptotic formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500.
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
FORMULA
| Expansion of psi(q) / chi(-q) = f(-q^2) / chi(-q)^2 = f(-q) / chi(-q)^3 = phi(-q) / chi(-q)^4 = phi(q) / chi(-q^2)^2 = f(-q^2)^2 / phi(-q) = f(-q)^4 / phi(-q)^3 = psi(q)^2 / f(-q^2) = chi(q)^2 * psi(q^2) = f(-q^2)^3 / f(-q)^2 in powers of q where f(), phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Feb 18 2006
Euler transform of period 2 sequence [ 2, -1, ...].
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k - 1))^2 = Product_{k>0} (1 - x^k) * (1 + x^k)^3.
a(n) = b(n)+b(n-1)+b(n-3)+b(n-6)+...+b(n-k*(k+1)/2)+..., where b() is A000009(). E.g. a(8) = b(8)+b(7)+b(5)+b(2) = 6+5+3+1 = 15. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 18 2004
|
|
|
EXAMPLE
| 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 12*x^7 + 15*x^8 + ...
q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 10*q^37 + 12*q^43 + 15*q^49 + ...
|
|
|
MATHEMATICA
| a[ n_] := SeriesCoefficient[ Product[ (1 - x^k) * (1 + x^k)^3, {k, n}], {x, 0, n}]
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 2, n, 2}] / Product[ 1 - x^k, {k, 1, n, 2}]^2, {x, 0, n}]
a[ n_] := With[ {t = Log[q]/(2 Pi I)}, SeriesCoefficient[ q^(-1/6) DedekindEta[ 2 t]^3 / DedekindEta[ t]^2, {q, 0, n}]]
|
|
|
PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A)^2, n))}
|
|
|
CROSSREFS
| Cf. A000009, A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 16 2010]
Sequence in context: A062436 A121269 A056219 * A138883 A107849 A053036
Adjacent sequences: A085137 A085138 A085139 * A085141 A085142 A085143
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Michael Somos, Jun 20 2003
|
| |
|
|
