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Expansion of f(x^2, x^3) * f(-x^4, -x^6) / f(-x^2) in powers of x where f(,) is the Ramanujan general theta function.
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%I #14 Feb 16 2025 08:33:26

%S 1,0,2,1,2,1,2,1,3,2,4,4,5,4,6,5,7,6,9,8,11,11,13,13,17,15,20,19,23,

%T 23,27,27,33,33,38,39,45,45,53,54,62,63,73,74,84,86,97,100,112,115,

%U 130,134,148,154,170,176,195,202,222,232,255,264,290,301,329

%N Expansion of f(x^2, x^3) * f(-x^4, -x^6) / f(-x^2) in powers of x where f(,) is the Ramanujan general theta function.

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

%C Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 9th equation.

%H G. C. Greubel, <a href="/A259551/b259551.txt">Table of n, a(n) for n = 0..1000</a>

%H G. E. Andrews, <a href="http://www.jstor.org/stable/2331943">An introduction to Ramanujan's "lost" notebook</a>, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 97, Equation (3.5)

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of f(x^2, x^3) * G(x^2) in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function.

%F Euler transform of period 10 sequence [ 0, 2, 1, -1, -1, -1, 1, 2, 0, -1, ...].

%F G.f.: Product_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 3)) * (1 + x^(5*k - 2)) / ((1 - x^(10*k - 8)) * (1 - x^(10*k - 2))).

%F G.f.: (Sum_{k in Z} x^(k*(5*k + 1)/2)) / (Product_{k in Z} 1 - x^abs(10*k + 2)). - _Michael Somos_, Jul 09 2015

%F (-1)^n * A053258(n) + A053266(n) = a(n) unless n=0. _Michael Somos_, Jul 09 2015

%F A259910(n) = 2*A255065(n) + a(n). _Michael Somos_, Jul 09 2015

%e G.f. = 1 + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 3*x^8 + 2*x^9 + ...

%e G.f. = 1/q + 2*q^239 + q^359 + 2*q^479 + q^599 + 2*q^719 + q^839 + ...

%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, -2, -1, 1, 1, 1, -1, -2, 0, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ QPochhammer[x^5] QPochhammer[ -x^2, x^5] QPochhammer[ -x^3, x^5] / (QPochhammer[ x^2, x^10] QPochhammer[ x^8, x^10]), {x, 0, n}];

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, -2, -1, 1, 1, 1, -1, -2, 0][k%10 + 1]), n))};

%Y Cf. A053258, A053266, A255065, A259910.

%K nonn,changed

%O 0,3

%A _Michael Somos_, Jun 30 2015