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A186741
Expansion of f(x^5, x^7) in powers of x where f(, ) is Ramanujan's general theta function.
1
1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
OFFSET
0,1
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Euler transform of period 24 sequence [ 0, 0, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, ...].
a(n) is the characteristic function of A036498. a(n) = max( 0, A010815(n)).
G.f.: Sum_{k in Z} x^(6*k^2 - k) = Product_{k>0} (1 + x^(12*k - 7)) * (1 + x^(12*k - 5)) * (1 - x^(12*k)).
Sum_{k=1..n} a(k) ~ sqrt(2*n/3). - Amiram Eldar, Jan 13 2024
EXAMPLE
G.f. = 1 + x^5 + x^7 + x^22 + x^26 + x^51 + x^57 + x^92 + x^100 + x^145 + ...
G.f. = q + q^121 + q^169 + q^529 + q^625 + q^1225 + q^1369 + q^2209 + q^2401 + ...
MATHEMATICA
a[n_]:= SeriesCoefficient[ QPochhammer[-q^5, q^12]*QPochhammer[-q^7, q^12] *QPochhammer[q^12, q^12], {q, 0, n}]; (* G. C. Greubel, dec 08 2017 *)
PROG
(PARI) {a(n) = my(m); if( !issquare( 24*n + 1, &m), 0, m%12 == 1 || m%12 == 11)};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 21 2012
STATUS
approved