A029838 Expansion of square root of q times normalized Hauptmodul for Gamma(4) in powers of q^8.

1, 1, -1, 0, 1, 0, -1, -1, 2, 1, -2, -1, 2, 1, -3, -1, 4, 2, -5, -2, 5, 2, -6, -3, 8, 4, -9, -4, 10, 4, -12, -6, 15, 7, -17, -7, 19, 8, -22, -10, 26, 12, -30, -13, 33, 14, -38, -17, 45, 21, -51, -22, 56, 24, -64, -29, 74, 33, -83, -36, 92, 40, -104, -46, 119, 53, -133, -58, 147, 63, -165, -73, 187, 83, -208, -90, 229, 99, -256

Offset

0,9

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).

Links

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

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.1),(9.3).

J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann. 318 (2000), no. 2, 255-275.

Michael Somos, Introduction to Ramanujan theta functions

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

Formula

Expansion of f(x) / f(-x^4) = phi(x) / psi(x) = psi(x) / psi(x^2) = phi(-x^2) / psi(-x) = chi(x) * chi(-x^2) = chi^2(x) * chi(-x) = chi^2(-x^2) / chi(-x) = (phi(x) / psi(x^2))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

Expansion of q^(1/8) * eta(q^2)^3 / (eta(q) * eta(q^4)^2) in powers of q.

Euler transform of period 4 sequence [ 1, -2, 1, 0, ...].

Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4 + v^4 - u^4*v^2. - Michael Somos, Mar 02 2006

Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = u^4 - v^4 - 4*u*v + u^3*v^3. - Michael Somos, Mar 02 2006

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) = 2 + w^2 - u^2*v*w. - Michael Somos, Mar 02 2006

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) = u2^2 + u6^2 - u1*u2*u3*u6. - Michael Somos, Mar 02 2006

G.f. A(x) satisfies A(x)^2 = (A(x^4) + 2*x / A(x^4)) / A(x^2). - Michael Somos, Mar 08 2004

G.f. A(x) satisfies A(x) = (A(x^2)^2+4*x/A(x^2)^2)^(1/4). - Joerg Arndt, Aug 06 2011

G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 + x^(2*k)) = (Sum_{k>0} x^((k^2 - k)/2)) / (Sum_{k>0} x^(k^2 - k)).

G.f.: 1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...))).

A082303(n) = (-1)^n a(n). Convolution square is A029839. Convolution inverse is A083365.

G.f.: 2 - 2/(1+Q(0)), where Q(k)= 1 + x^(k+1) + x^(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013

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} A109506(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 14 2017

abs(a(n)) ~ sqrt(sqrt(2) + (-1)^n) * exp(Pi*sqrt(n)/2^(3/2)) / (4*n^(3/4)). - Vaclav Kotesovec, Feb 07 2023

Example

G.f. = 1 + x - x^2 + x^4 - x^6 - x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + 2*x^12 + ...
G.f. = 1/q + q^7 - q^15 + q^31 - q^47 - q^55 + 2*q^63 + q^71 - 2*q^79 - q^87 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ q^2, q^4], {q, 0, n}]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, -q] / QPochhammer[ q^4, q^4], {q, 0, n}]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := SeriesCoefficient[ q^(1/8) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q], {q, 0, n}]; (* Michael Somos, Aug 20 2014 *)
(QPochhammer[-x, x^2, 1/2] + O[x]^100)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 + x^k)^(-(-1)^k), 1 + x * O(x^n)), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1 + if( j>1, x^(j-1))))); polcoeff(A[1, 1] / A[2, 1] + x * O(x^n), n))}; /* Michael Somos, Mar 02 2006 */
(PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A2 = subst(A, x, x^2); A = sqrt((A2 + 2  * x / A2) / A)); polcoeff(A, n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A) / eta(x^4 + A)^2, n))};

Crossrefs

Cf. A029839, A082303, A083365.

Sequence in context: A293224 A082303 A316384 * A213649 A242397 A023132

Adjacent sequences: A029835 A029836 A029837 * A029839 A029840 A029841

Keyword

sign

Author

N. J. A. Sloane

Status

approved