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A370019
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Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(1/3).
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7
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1, -4, -16, -48, -384, -2816, -24384, -206336, -1815552, -16189440, -146777856, -1346648064, -12487131136, -116810932224, -1101080592384, -10447586845696, -99706199973888, -956400813293568, -9215587975397376, -89158545637244928, -865730439117078528, -8433936444598677504
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OFFSET
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0,2
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COMMENTS
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The self-convolution cube equals A370018.
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^(4/3), where d = 10.39336299855957350315151176284030870108168399888... and c = -0.218294054014127126766352511836393819909572679... - Vaclav Kotesovec, Feb 24 2024
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EXAMPLE
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G.f.: A(x) = 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 - 146777856*x^10 + ...
RELATED SERIES.
The cube of the g.f. A(x) equals the g.f. A370018 which starts as
A(x)^3 = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
The reciprocal of the g.f. A(x) equals the g.f. of A370044, which begins
1/A(x) = 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 + ...
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PROG
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(PARI) {a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n))^(1/3);
polcoeff(H=A, n)}
for(n=0, 20, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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