|
| |
|
|
A096920
|
|
Expansion of q^(-1/12) * eta(q^2)^4 / (eta(q)^2 * eta(q^4)) in powers of q.
|
|
1
| |
|
|
1, 2, 1, 2, 3, 2, 4, 4, 4, 6, 7, 8, 8, 10, 11, 14, 16, 16, 20, 22, 24, 28, 32, 34, 39, 44, 48, 54, 60, 66, 73, 82, 88, 98, 108, 118, 132, 144, 156, 172, 188, 204, 224, 244, 265, 290, 316, 340, 372, 404, 436, 474, 513, 554, 600, 650, 700, 756, 816, 878, 948, 1022, 1096, 1182
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
FORMULA
| a(n) = b(n)+b(n-1)+b(n-3)+b(n-6)+...+b(n-k*(k+1)/2)+..., where b() is A000700(). E.g. a(8) = b(8)+b(7)+b(5)+b(2) = 2+1+1+0 = 4.
G.f.: Product_{k>0} (1 - x^(4*k)) * (1 + x^(2*k-1))^2. - Michael Somos, Mar 25 2008
Expansion of psi(q) * chi(q) = f(q) / chi(-q) = f(q)^2 / phi(-q^2) = phi(-q^2) / chi(-q)^2 = phi(q) / chi(-q^2) = psi(q)^2 / f(-q^4) = f(-q^4) * chi(q)^2 where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 2, -2, 2, -1, ...]. - Michael Somos Mar 25 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 12^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A138559.
|
|
|
EXAMPLE
| q + 2*q^13 + q^25 + 2*q^37 + 3*q^49 + 2*q^61 + 4*q^73 + 4*q^85 + 4*q^97 + ...
|
|
|
PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / eta(x + A)^2 / eta(x^4 + A), n))} /* Michael Somos Mar 25 2008 */
|
|
|
CROSSREFS
| Sequence in context: A192185 A080845 A029166 * A087154 A029839 A082304
Adjacent sequences: A096917 A096918 A096919 * A096921 A096922 A096923
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 18 2004
|
| |
|
|
