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A133738 Expansion of product of 3rd order mock theta function phi(q) and Ramanujan theta function f(-q) in powers of q.

%I A133738 #13 Feb 16 2025 08:33:06
%S A133738 1,0,-2,-2,2,2,-2,0,2,4,-2,-4,2,0,-2,-2,2,4,-4,-4,2,2,-2,0,4,4,0,-6,2,
%T A133738 0,-2,0,2,6,-4,-4,4,0,-4,-2,0,4,-2,-4,2,0,0,0,4,4,-2,-6,2,0,-6,2,2,8,
%U A133738 0,-4,2,0,0,0,2,2,-6,-4,2,0,-2,0,4,4,0,-6,2,-2
%N A133738 Expansion of product of 3rd order mock theta function phi(q) and Ramanujan theta function f(-q) in powers of q.
%C A133738 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A133738 G. C. Greubel, <a href="/A133738/b133738.txt">Table of n, a(n) for n = 0..1000</a>
%H A133738 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A133738 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A133738 G.f.: Sum_{k in Z} (-1)^k * x^(k*(3*k + 1)/2) * (1 + x^k) / (1 + x^(2*k)).
%F A133738 G.f.: ( Product_{k>0} 1 - x^k ) * ( 1 + Sum_{k>0} x^k^2 / ((1 + x^2) * (1 + x^4) * ... * (1 + x^(2*k))) ).
%F A133738 Convolution of A053250 and A010815.
%e A133738 G.f. = 1 - 2*x^2 - 2*x^3 + 2*x^4 + 2*x^5 - 2*x^6 + 2*x^8 + 4*x^9 - 2*x^10 + ...
%t A133738 a[ n_] := If[ n < 0, 0, With[ {m = Quotient[ Sqrt[24 n + 1] - 1, 6]}, SeriesCoefficient[ Sum[ (-1)^k x^(k (3 k + 1)/2) (1 + x^k) / (1 + x^(2 k)), {k, -m, m}], {x, 0, n}]]]; (* _Michael Somos_, Jul 26 2015 *)
%o A133738 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, (sqrtint(24*n + 1) - 1) \ 6, (-1)^k * x^(k*(3*k + 1)/2) * (1 + x^k) / (1 + x^(2*k)), x * O(x^n)), n))};
%o A133738 (PARI) {a(n) = my(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 + x^(2*k)) + x * O(x^(n - (k-1)^2)), 1) * eta(x + x * O(x^n)), n))};
%Y A133738 Cf. A010815, A053250.
%K A133738 sign
%O A133738 0,3
%A A133738 _Michael Somos_, Sep 22 2007