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A370338
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Expansion of Product_{n>=1} (1 - 3^(n-1)*x^n) * (1 + 3^(n-1)*x^n)^2.
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4
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1, 1, 2, 11, 24, 114, 297, 1224, 3240, 13230, 37017, 138510, 407754, 1469664, 4413366, 15717969, 47239200, 163408266, 511758000, 1719152586, 5348422224, 18083342907, 56672868240, 187301066040, 594207370746, 1947548449296, 6185182455792, 20263641256656, 64084643627283
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OFFSET
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0,3
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COMMENTS
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Compare to Product_{n>=1} (1 - 3^n*x^n) * (1 + 3^n*x^n)^2 = Sum_{n>=0} 3^(n*(n+1)/2) * x^(n*(n+1)/2).
In general, for d > 1, if g.f. = Product_{k>=1} (1 - d^(k-1)*x^k) * (1 + d^(k-1)*x^k)^2, then a(n) ~ c^(1/4) * d^(n + 3/2) * exp(2*sqrt(c*n)) / (2 * sqrt((d-1)*Pi) * (d+1) * n^(3/4)), where c = -2*polylog(2, -1/d) - polylog(2, 1/d). - Vaclav Kotesovec, Feb 26 2024
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LINKS
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FORMULA
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a(n) ~ c^(1/4) * 3^(n + 3/2) * exp(2*sqrt(c*n)) / (2^(7/2) * sqrt(Pi) * n^(3/4)), where c = -2*polylog(2,-1/3) - polylog(2,1/3) = 0.2518530229985534570173197... - Vaclav Kotesovec, Feb 26 2024
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 24*x^4 + 114*x^5 + 297*x^6 + 1224*x^7 + 3240*x^8 + 13230*x^9 + 37017*x^10 + 138510*x^11 + 407754*x^12 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 3*x^2)*(1 + 3*x^2)^2 * (1 - 9*x^3)*(1 + 9*x^3)^2 * (1 - 27*x^4)*(1 + 27*x^4)^2 * ... * (1 - 3^(n-1)*x^n)*(1 + 3^(n-1)*x^n)^2 * ...
Compare A(x) to the series that results from a similar infinite product:
(1 - 3*x)*(1 + 3*x)^2 * (1 - 9*x^2)*(1 + 9*x^2)^2 * (1 - 27*x^3)*(1 + 27*x^3)^2 * (1 - 81*x^4)*(1 + 81*x^4)^2 * ... = 1 + 3*x + 27*x^3 + 729*x^6 + 59049*x^10 + 14348907*x^15 + 10460353203*x^21 + 22876792454961*x^28 + ...
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PROG
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(PARI) {a(n) = polcoeff( prod(k=1, n, (1 - 3^(k-1)*x^k) * (1 + 3^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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