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A053274 Coefficients of the '6th-order' mock theta function gamma(q).

%I A053274 #28 May 23 2023 10:47:16
%S A053274 1,1,-1,0,2,-2,-1,3,-2,0,3,-4,-1,5,-3,-1,6,-6,-2,7,-6,0,9,-8,-3,11,-9,
%T A053274 -2,13,-13,-3,17,-12,-3,19,-18,-5,22,-19,-3,27,-24,-7,33,-26,-5,36,
%U A053274 -34,-9,44,-35,-9,51,-45,-11,58,-49,-9,68,-59,-16,78,-65,-15,88,-79,-19,104,-84,-19,117,-102,-26,133,-112,-24,152,-131
%N A053274 Coefficients of the '6th-order' mock theta function gamma(q).
%D A053274 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17
%H A053274 G. C. Greubel, <a href="/A053274/b053274.txt">Table of n, a(n) for n = 0..1000</a>
%H A053274 George E. Andrews and Dean Hickerson, <a href="https://doi.org/10.1016/0001-8708(91)90083-J">Ramanujan's "lost" notebook VII: The sixth order mock theta functions</a>, Advances in Mathematics, 89 (1991) 60-105.
%F A053274 G.f.: gamma(q) = Sum_{n >= 0} q^n^2/((1+q+q^2)(1+q^2+q^4)...(1+q^n+q^(2n))).
%F A053274 From _Seiichi Manyama_, May 23 2023: (Start)
%F A053274 a(n) = A328988(n) - A328989(n) for n > 0.
%F A053274 G.f.: 1 + (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1+x^k) * (1-x^k)^2 / (1+x^k+x^(2*k)). (End)
%t A053274 Series[Sum[q^n^2/Product[1+q^k+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
%o A053274 (PARI) a(n) = polcoeff(sum(k=0, 50, q^(k^2)/prod(j=1, k, 1+q^j+q^(2*j)), q*O(q^n)), n);
%o A053274 for(n=0,50, print1(a(n), ", ")) \\ _G. C. Greubel_, May 18 2018
%o A053274 (PARI) my(N=80, x='x+O('x^N)); Vec(1+1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1+x^k)*(1-x^k)^2/(1+x^k+x^(2*k)))) \\ _Seiichi Manyama_, May 23 2023
%Y A053274 Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053273.
%Y A053274 Cf. A328988, A328989.
%K A053274 sign,easy
%O A053274 0,5
%A A053274 _Dean Hickerson_, Dec 19 1999