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A370334 Expansion of [ Sum_{n>=0} 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2) ]^(1/3).

%I A370334 #8 Feb 27 2024 04:15:13
%S A370334 1,37,-1369,133632,-9888768,845367083,-78838949376,7721334144755,
%T A370334 -776624602305024,79868229118115328,-8362877755373222400,
%U A370334 888226662691859185152,-95442299152209579505105,10355840499178710443340288,-1132966823558169033184762368,124832961812953439236127605357
%N A370334 Expansion of [ Sum_{n>=0} 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2) ]^(1/3).
%H A370334 Paul D. Hanna, <a href="/A370334/b370334.txt">Table of n, a(n) for n = 0..300</a>
%F A370334 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A370334 (1) A(x)^3 = Sum_{n>=0} 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2).
%F A370334 (2) A(x)^3 = Product_{n>=1} (1 + 11^(n-1)*x^n) * (1 - 11^n*x^n) * (1 + 11^(n+1)*x^n), by the Jacobi triple product identity.
%F A370334 (3) A(x) = Product_{n>=1} F( 11^(n-1)*x^n ), where F(x) = (1 + 111*x - 1221*x^2 - 1331*x^3)^(1/3).
%F A370334 a(n) ~ (-1)^(n+1) * c * 11^(2*n) / n^(4/3), where c = 0.2588865455859866840901787578907966... - _Vaclav Kotesovec_, Feb 27 2024
%e A370334 G.f.: A(x) = 1 + 37*x - 1369*x^2 + 133632*x^3 - 9888768*x^4 + 845367083*x^5 - 78838949376*x^6 + 7721334144755*x^7 - 776624602305024*x^8 + ...
%e A370334 where the cube of g.f. A(x) yields the series
%e A370334 A(x)^3 = 1 + 111*x + 147631*x^3 + 2161452161*x^6 + 348104014265601*x^10 + 616687495357008127151*x^15 + 12017494675541950940487123311*x^21 + ... + 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2) + ...
%e A370334 The cube of g.f. A(x) also equals the infinite product
%e A370334 A(x)^3 = (1 + x)*(1 - 11*x)*(1 + 11^2*x) * (1 + 11*x^2)*(1 - 11^2*x^2)*(1 + 11^3*x^2) * (1 + 11^2*x^3)*(1 - 11^3*x^3)*(1 + 11^4*x^3) * (1 + 11^3*x^4)*(1 - 11^4*x^4)*(1 + 11^5*x^4) * ...
%o A370334 (PARI) {a(n) = my(A); A = prod(m=1, n+1, (1 + 11^(m-1)*x^m) * (1 - 11^m*x^m) * (1 + 11^(m+1)*x^m) +x*O(x^n))^(1/3); polcoeff(H=A, n)}
%o A370334 for(n=0, 30, print1(a(n), ", "))
%Y A370334 Cf. A370149, A370016, A370148, A370019, A370336.
%K A370334 sign
%O A370334 0,2
%A A370334 _Paul D. Hanna_, Feb 25 2024