A106507 G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)).

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

Offset

0,4

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The present entry gives 1/psi(q).

For various G.f. versions see the reciprocals of the ones given in A010054. - Wolfdieter Lang, Jul 05 2016

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 3rd equation, p. 41, 12th equation.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

C. Adiga, N. Anitha, T. Kim, Transformations of Ramanujan's Summation Formula and its Applications, arXiv:math/0501528 [math.NT], 2005. See page 5.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions, q-Pochhammer Symbol

Formula

Expansion of 1 / psi(x) in powers of x where psi() is a Ramanujan theta function, which is Jacobi's theta_2(0, sqrt(x))/(2*x^(1/8)) function. See, e.g., the Eric Weisstein link.

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(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^4 * (w^4 + 4*v^4) - v^6*w^2.

Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^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(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1^3*u6^2 + 3*u1^3*u2^2 - u2^3*u3*u6.

G.f.: Sum_{k>=0} a(k) * x^(8*k - 1) = 1 / (Sum_{k in Z} 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 the g.f. for A015128. - Michael Somos, Nov 01 2008

a(n) = (-1)^n * A006950(n). Convolution inverse of A010054.

Series reversion of A106336. - Michael Somos, May 10 2012

a(2*n) = A233758(n). a(2*n + 1) = - A233759(n). - Michael Somos, Nov 05 2015

G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(2*k)). - Michael Somos, Nov 08 2015

G.f.: (x; x^2)_{1/2}, where (a; q)_n is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 20 2016

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 08 2017

Example

G.f. = 1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 5*x^6 - 7*x^7 + 10*x^8 + ...
G.f. of B(q) =  A(q^8) / q = 1/q - q^7 + q^15 - 2*q^23 + 3*q^31 - 4*q^39 + 5*q^47 - 7*q^55 + ...

Mathematica

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^((-1)^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 28 2015 *)
a[ n_] := SeriesCoefficient[ 2 x^(1/8) / EllipticTheta[ 2, 0, x^(1/2)] , {x, 0, n}]; (* Michael Somos, Jun 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x] / QPochhammer[ x^4], {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)
(QPochhammer[x, x^2, 1/2] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

Prog

(PARI) {a(n) = my(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, A233758, A233759.

Sequence in context: A316720 A316721 A316722 * A006950 A052335 A193771

Adjacent sequences: A106504 A106505 A106506 * A106508 A106509 A106510

Keyword

sign,easy

Author

Michael Somos, May 04 2005

Extensions

Definition changed by N. J. A. Sloane, Aug 14 2007

Status

approved