A131962 Expansion of psi(x) * phi(-x^12) / chi(-x^4) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 2, 2, 1, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 3, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 0, 3, 1, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 2

Offset: 0

Author

Michael Somos, Aug 02 2007

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^7 + q^31 + q^79 + q^103 + q^127 + q^151 + q^175 + q^199 + q^223 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A093766, A123484.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 7}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] EllipticTheta[ 4, 0, x^12] QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 4, 0, x^12] QPochhammer[ -x^4, x^4], {x, 0, n}]; (* Michael Somos, Nov 06 2015 *)

PARI

{a(n) = if( n<0, 0, n = 24*n + 7; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^24 + A)), n))};

Formula

Expansion of q^(-7/24) * eta(q^2)^2 * eta(q^8) * eta(q^12)^2/( eta(q) * eta(q^4) * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [ 1, -1, 1, 0, 1, -1, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, -1, 1, 0, 1, -1, 1, -2, ...].

a(25*n + 7) = a(n). a(25*n + 2) = a(25*n + 12) = a(25*n + 17) = a(25*n + 22) = 0.

2 * a(n) = A123484(24*n + 7).

Expansion of chi(x) * f(-x^8) * phi(-x^12) in powers of x where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Nov 06 2015

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Jan 20 2025