%I #9 Feb 25 2024 05:01:07

%S 1,-12,0,176,0,0,-2752,0,0,0,43776,0,0,0,0,-699392,0,0,0,0,0,11186176,

%T 0,0,0,0,0,0,-178962432,0,0,0,0,0,0,0,2863333376,0,0,0,0,0,0,0,0,

%U -45813071872,0,0,0,0,0,0,0,0,0,733008101376,0,0,0,0,0,0,0,0,0,0,-11728125427712

%N Expansion of Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).

%C Equals the self-convolution cube of A370019.

%e G.f.: A(x) = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 - 178962432*x^28 + 2863333376*x^36 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...

%e RELATED SERIES.

%e The cube root of g.f. A(x) is an integer series starting as

%e A(x)^(1/3) = 1 - 4*x - 16*x^2 - 48*x^3 - 384*x^4 - 2816*x^5 - 24384*x^6 - 206336*x^7 - 1815552*x^8 - 16189440*x^9 + ... + A370019(n)*x^n + ...

%e Also,

%e A(x)^(1/6) = 1 - 2*x - 10*x^2 - 44*x^3 - 330*x^4 - 2508*x^5 - 21476*x^6 - 185720*x^7 - 1658778*x^8 - 15042060*x^9 - 138464620*x^10 + ...

%e The expansion of 1/A(x) begins

%e 1/A(x) = 1 + 12*x + 144*x^2 + 1552*x^3 + 16512*x^4 + 172800*x^5 + 1803200*x^6 + 18765312*x^7 + 195167232*x^8 + 2028914688*x^9 + ... + A370045(n)*x^n + ...

%e Further,

%e 1/A(x)^(1/3) = 1 + 4*x + 32*x^2 + 240*x^3 + 2048*x^4 + 17920*x^5 + 163904*x^6 + 1526784*x^7 + 14473216*x^8 + 138743808*x^9 + ... + A370044(n)*x^n + ...

%o (PARI) {a(n) = my(A);

%o A = sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n));

%o polcoeff(H=A, n)}

%o for(n=0, 66, print1(a(n), ", "))

%Y Cf. A370019, A370045, A370044, A370015, A370335.

%K sign

%O 0,2

%A _Paul D. Hanna_, Feb 23 2024