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A301310
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G.f.: Sum_{n>=0} 2^n * (1+x)^(n^2) / 3^(n+1).
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6
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1, 10, 360, 21840, 1857660, 203258160, 27188330400, 4298562686880, 784233322674120, 162161079972261480, 37477229047577953920, 9573364920705562944000, 2678416661190852872256960, 814535089079749159186189440, 267528376262254011309768677760, 94377360018309519999410315205120, 35590366640535756970223476489499280, 14287353028920891078189826021459809120
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OFFSET
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0,2
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COMMENTS
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Is there a finite expression for the terms of this sequence?
a(n) is divisible by 10 for n>0 (conjecture).
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LINKS
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FORMULA
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G.f.: 1/(3 - 2*q/(1 - 2*q*(q^2-1)/(3 - 2*q^5/(1 - 2*q^3*(q^4-1)/(3 - 2*q^9/(1 - 2*q^5*(q^6-1)/(3 - 2*q^13/(1 - 2*q^7*(q^8-1)/(3 - ...))))))))) where q = (1+x), a continued fraction due to a partial elliptic theta function identity.
G.f.: Sum_{n>=0} 2^n/3^(n+1) * (1+x)^n * Product_{k=1..n} (3 - 2*(1+x)^(4*k-3)) / (3 - 2*(1+x)^(4*k-1)), due to a q-series identity.
a(n) = Sum_{k>=0} 2^k * binomial(k^2, n) / 3^(k+1).
a(n) ~ 2^(2*n + 1/2 + log(3/2)/8) * n^n / (3^(1 + log(3/2)/8) * exp(n) * (log(3/2))^(2*n + 1)). - Vaclav Kotesovec, Mar 21 2018
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EXAMPLE
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G.f.: A(x) = 1 + 10*x + 360*x^2 + 21840*x^3 + 1857660*x^4 + 203258160*x^5 + 27188330400*x^6 + 4298562686880*x^7 + 784233322674120*x^8 + ...
such that
A(x) = 1/3 + 2*(1+x)/3^2 + 2^2*(1+x)^4/3^3 + 2^3*(1+x)^9/3^4 + 2^4*(1+x)^16/3^5 + 2^5*(1+x)^25/3^6 + 2^6*(1+x)^36/3^7 + 2^7*(1+x)^49/3^8 + 2^8*(1+x)^64/3^9 + ...
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MATHEMATICA
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Table[Sum[StirlingS1[n, j] * HurwitzLerchPhi[2/3, -2*j, 0]/3, {j, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Mar 21 2018 *)
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PROG
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(PARI) /* Continued fraction expression: */
{a(n) = my(CF=1, q = 1+x +x*O(x^n)); for(k=0, n, CF = 1/(3 - 2*q^(4*n-4*k+1)/(1 - 2*q^(2*n-2*k+1)*(q^(2*n-2*k+2) - 1)*CF)) ); polcoeff(CF, 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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