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A260971 Expansion of phi_0(-q) in powers of q where phi_0() is a 5th-order mock theta function.

%I A260971 #13 Feb 02 2021 20:24:34
%S A260971 1,-1,1,0,1,-1,0,-1,1,-1,1,0,1,-1,1,-1,1,-2,1,-1,1,-1,1,-1,2,-2,2,-1,
%T A260971 2,-2,1,-2,2,-2,2,-2,2,-3,2,-2,3,-3,3,-2,3,-3,3,-3,3,-4,4,-3,4,-4,3,
%U A260971 -4,4,-5,4,-4,5,-5,5,-5,6,-6,5,-5,6,-6,6,-6,7,-7,7
%N A260971 Expansion of phi_0(-q) in powers of q where phi_0() is a 5th-order mock theta function.
%H A260971 G. C. Greubel, <a href="/A260971/b260971.txt">Table of n, a(n) for n = 0..2500</a>
%F A260971 G.f.: Sum_{k >= 0}  (-x)^n^2 * (1 - x) * (1 - x^3) * ... * (1 - x^(2*k-1)).
%F A260971 a(n) = (-1)^n * A053258(n) = 2 * A053264(n) - A053262(n).
%F A260971 a(n) ~ (-1)^n * sqrt(phi) * exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 15 2019
%e A260971 G.f. = 1 - x + x^2 + x^4 - x^5 - x^7 + x^8 - x^9 + x^10 + x^12 - x^13 + ...
%e A260971 G.f. = q^-1 - q^119 + q^239 + q^479 - q^599 - q^839 + q^959 - q^1079 + ...
%t A260971 a[n_]:= SeriesCoefficient[Sum[(-x)^( k^2)*Product[1 - x^(2*j - 1), {j, 1, k}], {k, 0, Sqrt[n]}], {x, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Aug 01 2018 *)
%o A260971 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), (-x)^k^2 * prod(i=1, k, 1 - x^(2*i - 1), 1 + x * O(x^(n - k^2)))), n))};
%Y A260971 Cf. A053258, A053262, A053264.
%K A260971 sign
%O A260971 0,18
%A A260971 _Michael Somos_, Aug 06 2015