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URL: https://oeis.org/A214263

⇱ A214263 - OEIS

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A214263
Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.
10
1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
OFFSET
0,1
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Characteristic function of A074377: a(n) = 1 if and only if n is in A074377.
FORMULA
Euler transform of period 16 sequence [ 1, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 1, -1, ...].
G.f.: f(x, x^7) = sum_{k in Z} x^(4*k^2 - 3*k).
a(n) = A010054(2*n + 1) = A115359(2*n).
Sum_{k=1..n} a(k) ~ sqrt(n). - Amiram Eldar, Jan 13 2024
EXAMPLE
G.f. = 1 + x + x^7 + x^10 + x^22 + x^27 + x^45 + x^52 + x^76 + x^85 + x^115 + ...
G.f. = q^9 + q^25 + q^121 + q^169 + q^361 + q^441 + q^729 + q^841 + q^1225 + ...
MATHEMATICA
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; Table[SeriesCoefficient[f[q, q^7], {q, 0, n}], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)
PROG
(PARI) {a(n) = issquare(16*n + 9)};
CROSSREFS
A000122, A080995, A010054, A133100, A089801 have g.f. of f(x,x^k) for k=1..5.
Sequence in context: A374121 A292438 A244525 * A263428 A016355 A016402
KEYWORD
nonn,easy
AUTHOR
Michael Somos and Omar E. Pol, Jul 09 2012
STATUS
approved