This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370016 #26 Mar 07 2024 11:56:34
%S A370016 1,1,-1,9,-18,44,-54,350,-1359,3789,-9585,42489,-163900,543474,
%T A370016 -1933092,7499404,-27668718,100329714,-371138346,1394575578,
%U A370016 -5236658316,19587163968,-73536845444,278088068628,-1052804678958,3985553554074,-15132118280498,57617112474306,-219680808219216
%N A370016 Expansion of A(x) = [ Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) ]^(1/3).
%C A370016 Equals the self-convolution cube root of A370015.
%H A370016 Paul D. Hanna, <a href="/A370016/b370016.txt">Table of n, a(n) for n = 0..630</a>
%F A370016 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A370016 (1) A(x)^3 = Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2).
%F A370016 (2) A(x)^3 = Product_{n>=1} (1 + 2^(n-1)*x^n) * (1 - 2^n*x^n) * (1 + 2^(n+1)*x^n), by the Jacobi triple product identity.
%F A370016 (3) A(x) = Product_{n>=1} F( 2^(n-1)*x^n ), where F(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3).
%F A370016 a(n) ~ c * (-1)^(n+1) * 4^n / n^(4/3), where c = 0.260357663494676514371316... - _Vaclav Kotesovec_, Feb 23 2024
%F A370016 Radius of convergence r = 1/4 (from Vaclav Kotesovec's formula) and A(r) = ( Sum_{n>=0} (1/2^n + 2*2^n)/3 * 1/2^(n*(n+1)/2) )^(1/3) = ( Product_{n>=1} (1 + 1/2^n)*(1 - 1/4^(n+1)) )^(1/3) = 1.298389210904220681888354941631161162163... - _Paul D. Hanna_, Mar 07 2024
%e A370016 G.f.: A(x) = 1 + x - x^2 + 9*x^3 - 18*x^4 + 44*x^5 - 54*x^6 + 350*x^7 - 1359*x^8 + 3789*x^9 - 9585*x^10 + 42489*x^11 - 163900*x^12 + 543474*x^13 - 1933092*x^14 + 7499404*x^15 + ...
%e A370016 where the cube of g.f. A(x) yields the series
%e A370016 A(x)^3 = 1 + 3*x + 22*x^3 + 344*x^6 + 10944*x^10 + 699392*x^15 + 89489408*x^21 + 22907191296*x^28 + 11728213508096*x^36 + ... + 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) + ...
%e A370016 The cube of g.f. A(x) also equals the infinite product
%e A370016 A(x)^3 = (1 + x)*(1 - 2*x)*(1 + 2^2*x) * (1 + 2*x^2)*(1 - 2^2*x^2)*(1 + 2^3*x^2) * (1 + 2^2*x^3)*(1 - 2^3*x^3)*(1 + 2^4*x^3) * (1 + 2^3*x^4)*(1 - 2^4*x^4)*(1 + 2^5*x^4) * ...
%e A370016 Equivalently,
%e A370016 A(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) * (1 + 6*x^2 - 24*x^4 - 64*x^6)^(1/3) * (1 + 12*x^3 - 96*x^6 - 512*x^9)^(1/3) * (1 + 24*x^4 - 384*x^8 - 4096*x^12)^(1/3) * (1 + 48*x^5 - 1536*x^10 - 32768*x^15)^(1/3) * ...
%e A370016 where
%e A370016 (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) = 1 + x - 3*x^2 + 3*x^3 - 12*x^4 + 30*x^5 - 102*x^6 + 318*x^7 - 1083*x^8 + 3657*x^9 + ... + A370145(n)*x^n + ...
%t A370016 Round[CoefficientList[Series[QPochhammer[-2, 2*x]^(1/3) * QPochhammer[-1/2, 2*x]^(1/3) * QPochhammer[2*x]^(1/3)*2^(1/3)/3^(2/3), {x, 0, 30}], x]] (* _Vaclav Kotesovec_, Feb 23 2024 *)
%o A370016 (PARI) {a(n) = my(A);
%o A370016 A = prod(m=1, n+1, (1 + 2^(m-1)*x^m) * (1 - 2^m*x^m) * (1 + 2^(m+1)*x^m) +x*O(x^n))^(1/3);
%o A370016 polcoeff(H=A, n)}
%o A370016 for(n=0, 30, print1(a(n), ", "))
%Y A370016 Cf. A370015, A370145, A370148, A370334, A370019, A370336.
%Y A370016 Cf. A304961, A370716, A370765.
%K A370016 sign
%O A370016 0,4
%A A370016 _Paul D. Hanna_, Feb 22 2024