|
| |
|
|
A106507
|
|
G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)).
|
|
2
|
|
|
|
1, -1, 1, -2, 3, -4, 5, -7, 10, -13, 16, -21, 28, -35, 43, -55, 70, -86, 105, -130, 161, -196, 236, -287, 350, -420, 501, -602, 722, -858, 1016, -1206, 1431, -1687, 1981, -2331, 2741, -3206, 3740, -4368, 5096, -5922, 6868, -7967, 9233, -10670, 12306, -14193, 16357, -18803, 21581
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
|
|
|
REFERENCES
|
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41
|
|
|
LINKS
|
Table of n, a(n) for n=0..50.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
C. Adiga, N. Anitha, T. Kim, Transformations of Ramanujan's Summation Formula and its Applications, See page 5
|
|
|
FORMULA
|
Expansion of 1 / psi(x) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(1/8) * eta(q) / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -1, 1, ...].
Given g.f. A(x), then B(x) = A(x^8) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = u^4 * (w^4 + 4*v^4) - v^6*w^2.
Given g.f. A(x), then B(x) = A(x^8) / x satisfies 0 = f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = u1*u2*u6^3 + u2^2*u3^3 - u3^3*u6^2.
Given g.f. A(x), then B(x) = A(x^8) / x satisfies 0 = f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = u1^3*u6^2 + 3*u1^3*u2^2 - u2^3*u3*u6.
Sum_{k>=0} a(k) * x^(8*k - 1) = 1 / (Sum_{k} x^((4k + 1)^2)).
G.f.: 1 / (1 + x + x^3 + x^6 + ...) = 1 - x * (1 - x) / (1 - x^2)^2 + x^4 * (1 - x) * (1 - x^2) / ((1 - x^2)^2 * (1 - x^4)^2) + ... [Ramanujan] (Michael Somos, Jul 21 2008)
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(1/2) (t / i)^(-1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A015128. - Michael Somos, Nov 01 2008
a(n) = (-1)^n * A006950(n). Convolution inverse of A010054.
Reversion of A106336. - Michael Somos, May 10 2012
|
|
|
EXAMPLE
|
1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 5*x^6 - 7*x^7 + 10*x^8 +...
1/q - q^7 + q^15 - 2*q^23 + 3*q^31 - 4*q^39 + 5*q^47 - 7*q^55 +...
|
|
|
PROG
|
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A)^2, n))}
|
|
|
CROSSREFS
|
Cf. A006950, A010054, A106336.
Sequence in context: A014670 A036034 A006950 * A052335 A193771 A160333
Adjacent sequences: A106504 A106505 A106506 * A106508 A106509 A106510
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
Michael Somos, May 04 2005
|
|
|
EXTENSIONS
|
Definition changed by N. J. A. Sloane, Aug 14 2007
|
|
|
STATUS
|
approved
|
| |
|
|
