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
In Jacobi, "New Foundations ..." English translation page 99 the q-series A is the square of the generating function of this sequence. - Michael Somos, Sep 23 2025
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 3rd equation, p. 41, 12th equation.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, and T. Kim, Transformations of Ramanujan's Summation Formula and its Applications, arXiv:math/0501528 [math.NT], 2005. See page 5.
C. G. J. Jacobi, New Foundations of the Theory of Elliptic Functions (translation by Alexander Aycock).
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
Series reversion of A106336. - Michael Somos, May 10 2012
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
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = exp(-Pi/8) * 2^(5/8) * Gamma(3/4) / Pi^(1/4) = A388503. - Simon Plouffe, Sep 17 2025
Convolution square is A274621. - Michael Somos, Sep 23 2025
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 *)
a[ n_] := PowerExpand@SeriesCoefficient[ QPochhammer[ x, x^2, 1/2], {x, 0, n}]; (* Michael Somos, Sep 23 2025 *)
a[0] := 1; a[n_]/; n>=1 := a[n] = -Sum[a[n-k/2(k+1)], {k, 1, Sqrt[2 n]}]; a[n_] := 0; Table[a[n], {n, 0, 50}] (* Oliver Seipel, Mar 01 2026 *)
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
KEYWORD
sign,easy
AUTHOR
Michael Somos, May 04 2005
EXTENSIONS
Definition changed by N. J. A. Sloane, Aug 14 2007
STATUS
approved
