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%I #5 Feb 24 2024 11:04:18
%S 1,15,0,275,0,0,5375,0,0,0,106875,0,0,0,0,2134375,0,0,0,0,0,42671875,
%T 0,0,0,0,0,0,853359375,0,0,0,0,0,0,0,17066796875,0,0,0,0,0,0,0,0,
%U 341333984375,0,0,0,0,0,0,0,0,0,6826669921875,0,0,0,0,0,0,0,0,0,0,136533349609375
%N Expansion of Sum_{n>=0} 5^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).
%C Equals the self-convolution cube of A370336.
%e G.f.: A(x) = 1 + 15*x + 275*x^3 + 5375*x^6 + 106875*x^10 + 2134375*x^15 + 42671875*x^21 + 853359375*x^28 + 17066796875*x^36 + 341333984375*x^45 + ...
%e RELATED SERIES.
%e The cube root of the g.f. A(x) is an integer series starting as
%e A(x)^(1/3) = 1 + 5*x - 25*x^2 + 300*x^3 - 3000*x^4 + 34375*x^5 - 426750*x^6 + 5539375*x^7 - 73968750*x^8 + 1010175000*x^9 + ... + A370336(n)*x^n + ...
%o (PARI) {a(n) = my(A);
%o A = sum(m=0, sqrtint(2*n+1), 5^m*(2*4^m + 1)/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. A370015.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 23 2024
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