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A370764
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a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k))^(1/2).
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2
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1, 10, 62, 1620, 6966, 157580, 1284012, 19189160, 73908774, 2233414620, 9656822916, 287668788120, -324007115716, 40151699854200, -199460032590312, 7130611518222160, -64971542557275642, 1292318115470489340, -15433712240157937260, 265667290368470451000, -3624776372747687578668
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OFFSET
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0,2
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COMMENTS
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In general, if d > 1 and g.f. = Product_{k>=1} ((1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k))^(1/2), then a(n) ~ (-1)^(n+1) * QPochhammer(-1/d) * d^(2*n) / (2*sqrt((1 + 1/d)*Pi) * n^(3/2)).
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LINKS
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FORMULA
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G.f.: Product_{k>=1} ((1 + 2^(3*k+1)*x^k) * (1 + 2^(3*k-1)*x^k))^(1/2).
a(n) ~ (-1)^(n+1) * c * 16^n / n^(3/2), where c = QPochhammer(-1/2) / sqrt(6*Pi) = 0.278865402428524528968820654198674...
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MATHEMATICA
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nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*x^k)*(1+2^(k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[(1+2^(3*k+1)*x^k)*(1+2^(3*k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, x]*QPochhammer[-1/2, x])^(1/2)/3, {x, 0, nmax}], x] * 8^Range[0, nmax]
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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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