cp's OEIS Frontend

A370015 Expansion of A(x) = Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2).

%I A370015 #9 Mar 29 2025 03:26:58
%S A370015 1,3,0,22,0,0,344,0,0,0,10944,0,0,0,0,699392,0,0,0,0,0,89489408,0,0,0,
%T A370015 0,0,0,22907191296,0,0,0,0,0,0,0,11728213508096,0,0,0,0,0,0,0,0,
%U A370015 12009621912813568,0,0,0,0,0,0,0,0,0,24595670493070098432,0,0,0,0,0,0,0,0,0,0,100743830310818104213504
%N A370015 Expansion of A(x) = Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2).
%C A370015 Equals the self-convolution cube of A370016.
%F A370015 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A370015 (1) A(x) = Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2).
%F A370015 (2) A(x) = 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 A370015 (3) A(x) = Product_{n>=1} F( 2^(n-1)*x^n ), where F(x) = (1 + 3*x - 6*x^2 - 8*x^3).
%e A370015 G.f.: A(x) = 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 A370015 The g.f. A(x) equals the infinite product
%e A370015 A(x) = (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 A370015 equivalently,
%e A370015 A(x) = (1 + 3*x - 6*x^2 - 8*x^3) * (1 + 6*x^2 - 24*x^4 - 64*x^6) * (1 + 12*x^3 - 96*x^6 - 512*x^9) * (1 + 24*x^4 - 384*x^8 - 4096*x^12 ) * (1 + 48*x^5 - 1536*x^10 - 32768*x^15) * ...
%e A370015 Notice that the cube root of A(x) yields an integer series
%e A370015 A(x)^(1/3) = 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 + ... + A370016(n)*x^n + ...
%o A370015 (PARI) {a(n) = my(A);
%o A370015 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));
%o A370015 polcoeff(H=A, n)}
%o A370015 for(n=0, 66, print1(a(n), ", "))
%Y A370015 Cf. A370016.
%K A370015 nonn
%O A370015 0,2
%A A370015 _Paul D. Hanna_, Feb 22 2024