0,4

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Expansion of q^(-1/24) * eta(q) * eta(q^6)^5 / (eta(q^3)^2 * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ -1, -1, 1, -1, -1, -4, -1, -1, 1, -1, -1, -2, ...].

a(n) = b(24*n + 1) where b() is multiplicative with b(2^e) = 0, b(3^e) = I^e, b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 23 (mod 24), b(p^e) = I^e * (1 + (-1)^e) / 2 if p == 5, 7, 11 (mod 24), b(p^e) = (e+1) * s^e where s = Kronecker( 12, x) if p == 1 (mod 24), b(p^e) = (e+1) * s^e where s = -I * Kronecker( 12, x) if p == 19 (mod 24), where p = x^2 + 18 * y^2.

Given the 3-section A = A0 + A1 + A2, then 0 = A0 * A1^2 + A1 * A2^2 + A2 * A0^2.

G.f.: (Product_{k>0} 1 - x^k) * (Sum_{k in Z} x^(3*k^2)).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/8) * exp(Pi / 24) * 2^(1/8) * 3^(1/2) * Gamma(2/3) * Gamma(11/12)^2 * Gamma(7/12)^3 * (1+3^(1/2)) / Gamma(3/4)^7 = A388797. - Simon Plouffe, Sep 18 2025

G.f. = 1 - x - x^2 + 2*x^3 - 2*x^4 - x^5 + x^7 + 2*x^8 + 2*x^10 + x^12 - 2*x^13 + ...

G.f. = q - q^25 - q^49 + 2*q^73 - 2*q^97 - q^121 + q^169 + 2*q^193 + 2*q^241 + ...

a[ n_] := Module[ {m = 24 n + 1, i}, If[ n < 0, 0, Sum[ i = Sqrt[ m - 72 j^2]; If[ IntegerQ @ i, (1 + Boole[ j > 0]) JacobiSymbol[ 12, i], 0], {j, 0, Sqrt[ m / 72]}]]];

a[ n_] := SeriesCoefficient[ QPochhammer[ q] EllipticTheta[ 3, 0, q^3], {q, 0, n}];

(PARI) {a(n) = my(m, i); if( n<0, 0, m = 24*n + 1; sum( j=0, sqrtint( m \ 72), if( issquare( m - 72 * j^2, &i), (1 + (j>0)) * kronecker( 12, i), 0)))};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^5 / (eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

(PARI) {a(n) = my(A, p, e, i); if( n<0, 0, n = 24*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; I^e * if( p<5, p>2, if( p%24 == 1 || p%24 == 19, forstep( j = 3 * (p%24>1), sqrtint( p \ 2), 6, if( issquare( p - 2*j^2, &i), break)); (- kronecker( 12, i) * if( p%24>1, 1, I))^e * (e+1), if( e%2, 0, I^((p%24>12)*e)))))))};

Sequence in context: A180010 A287401 A003406 * A107063 A290453 A108423

Adjacent sequences: A226286 A226287 A226288 * A226290 A226291 A226292

Michael Somos, Jun 02 2013

approved