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

0,11

Comments

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

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.

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A093766, A123484.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A154326 A330460 A027186 * A302236 A262929 A226862

Adjacent sequences: A131959 A131960 A131961 * A131963 A131964 A131965

Keyword

nonn

Author

Michael Somos, Aug 02 2007

Status

approved