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A143063 Expansion of the product of a false theta function and a Ramanujan theta function in powers of x.

%I A143063 #12 Feb 16 2025 08:33:08
%S A143063 1,0,0,2,0,0,0,2,2,0,2,2,2,0,2,4,4,2,4,6,4,4,4,8,8,6,8,12,10,10,12,16,
%T A143063 16,14,18,22,22,20,24,30,32,30,36,42,42,42,48,56,60,58,66,76,78,80,88,
%U A143063 102,106,108,120,134,140,144,158,178,186,192,210,232,242,252,272,300
%N A143063 Expansion of the product of a false theta function and a Ramanujan theta function in powers of x.
%C A143063 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A143063 G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 235, Entry 9.4.8
%D A143063 S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 14th equation.
%H A143063 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A143063 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A143063 G.f.: (1 - x + x^2 - x^5 + x^7 - x^12 + x^15 - ...) * (1 + x) * (1 + x^3) * (1 + x^5) * (1 + x^7) * ... [Ramanujan]
%F A143063 G.f.: 1 + 2 * x^3 / (1 - x^4) + 2 * x^8 / ((1 - x^2) * (1 - x^8)) + 2 * x^15 / ((1 - x^2) * (1 - x^4) * (1 - x^12)) + 2 * x^24 / ((1 - x^2) * (1 - x^4) * (1 - x^6) * (1 - x^16)) + ... [Ramanujan]
%F A143063 a(n) = 2 * A027348(n) unless n=0. Convolution of A143062 and A000700.
%e A143063 G.f. = 1 + 2*x^3 + 2*x^7 + 2*x^8 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^14 + 4*x^15 + ...
%o A143063 (PARI) {a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=0, n, if( issquare( 24*k + 1, &m), (-1)^(m \ 3) * x^k ), A) / eta(x + A) * eta(x^2 + A)^2 / eta(x^4 + A), n))};
%o A143063 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, sqrtint(n+1) - 1, x^(k^2 + 2*k) / (1 - x^(4*k)) / prod(j=1, k-1, 1 - x^(2*j), 1 + O(x^(n + 1 - k^2 - 2*k)))), n))};
%Y A143063 Cf. A000700, A027348, A143062.
%K A143063 nonn
%O A143063 0,4
%A A143063 _Michael Somos_, Jul 21 2008