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allocated for Paul D. Hanna

Expansion of Sum_{n>=0} 5^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).

1, 15, 0, 275, 0, 0, 5375, 0, 0, 0, 106875, 0, 0, 0, 0, 2134375, 0, 0, 0, 0, 0, 42671875, 0, 0, 0, 0, 0, 0, 853359375, 0, 0, 0, 0, 0, 0, 0, 17066796875, 0, 0, 0, 0, 0, 0, 0, 0, 341333984375, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6826669921875, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 136533349609375

0,2

Equals the self-convolution cube of A370336.

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 + ...

RELATED SERIES.

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

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 + ...

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

A = sum(m=0, sqrtint(2*n+1), 5^m*(2*4^m + 1)/3 * x^(m*(m+1)/2) +x*O(x^n));

polcoeff(H=A, n)}

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

Cf. A370015.

allocated

nonn

Paul D. Hanna, Feb 23 2024

approved

editing

allocated for Paul D. Hanna

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