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 A274179 Expansion of f(x^1, x^6) in powers of x where f() is Ramanujan's general theta function. 5
 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 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, 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, 1, 0, 0, 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, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA G.f.: f(x, x^6) = Sum_{k in Z} x^((7k^2 + 5k)/2). G.f.: Product_{k>0} (1 + x^(7*k-6)) * (1 + x^(7*k-1)) * (1 - x^(7*k)). - Michael Somos, Jan 28 2017 a(3*n + 2) = a(5*n + 2) = a(5*n + 3) = 0. - Michael Somos, Jan 28 2017 a(n) = abs(A232714(n)). - Michael Somos, Jan 28 2017 EXAMPLE G.f. = 1 + x + x^6 + x^9 + x^19 + x^24 + x^39 + x^46 + x^66 + x^75 + x^100 + ... G.f. = q^25 + q^61 + q^241 + q^349 + q^709 + q^889 + q^1429 + q^1681 + q^2401 + ... MATHEMATICA a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt @ n]; (* Michael Somos, Jan 28 2017 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^7] QPochhammer[ -x^6, x^7] QPochhammer[ x^7], {x, 0, n}]; (* Michael Somos, Jan 28 2017 *) a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 1, -1, 1}[[Mod[k, 14, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Jan 28 2017 *) PROG (PARI) {a(n) = issquare(56*n + 25)}; /* Michael Somos, Jan 28 2017 */ (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[1, -1, 1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 1, -1][k%14 + 1], 1 + x * O(x^n)), n))}; /* Michael Somos, Jan 28 2017 */ CROSSREFS Cf. A232714. Cf. f(x, x^k): A080995 (k=2), A010054 (k=3), A133100 (k=4), A089801(k=5), this sequence (k=6), A214263 (k=7). Sequence in context: A362130 A361018 A232714 * A194675 A117964 A179020 Adjacent sequences: A274176 A274177 A274178 * A274180 A274181 A274182 KEYWORD nonn AUTHOR Seiichi Manyama, Jun 12 2016 STATUS approved

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Last modified June 21 20:48 EDT 2024. Contains 373559 sequences. (Running on oeis4.)