A301311 G.f.: Sum_{n>=0} 2^n * (1-x)^(-n^2) / 3^(n+1).

1, 10, 370, 22570, 1924270, 210821290, 28223418010, 4464779024650, 814901395935550, 168556843188104050, 38965275697707264970, 9955529477371346769010, 2785811940289987110605590, 847316256984037311888049090, 278329013908504193489288029090, 98197864581209379156337136722690, 37034491818759647215732974465421990, 14868275488492647637389364332301206490

Offset

0,2

Comments

Is there a finite expression for the terms of this sequence?

a(n) is divisible by 10 for n>0 (conjecture).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

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/(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*(1-x)^(4*k-3) - 2) / (3*(1-x)^(4*k-1) - 2), due to a q-series identity.

a(n) = Sum_{k>=0} 2^k * binomial(k^2 + n-1, n) / 3^(k+1).

a(n) ~ 2^(2*n + 1/2 - log(3/2)/8) * 3^(log(3/2)/8 - 1) * n^n / (exp(n) * (log(3/2))^(2*n + 1)). - Vaclav Kotesovec, Mar 21 2018

Example

G.f.: A(x) = 1 + 10*x + 370*x^2 + 22570*x^3 + 1924270*x^4 + 210821290*x^5 + 28223418010*x^6 + 4464779024650*x^7 + 814901395935550*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 + ...

Prog

(PARI) /* Continued fraction expression: */
{a(n) = my(CF=1, q = 1/(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), ", "))

Crossrefs

Cf. A104209, A301310.

Sequence in context: A220638 A119547 A117797 * A291530 A358801 A117312

Adjacent sequences: A301308 A301309 A301310 * A301312 A301313 A301314

Keyword

nonn

Author

Paul D. Hanna, Mar 18 2018

Status

approved