A338278 - OEIS

%I #14 Oct 24 2020 04:47:28

%S 1,3,30,507,12027,367167,13706049,604806546,30798675507,1777651129164,

%T 114681681511572,8177632124654658,638681988964923723,

%U 54220733932072259388,4971400927192150334778,489591085640900144694105,51541568078546978411976714,5776152891297758939283199518

%N G.f.: Sum_{n>=0} ( Product_{k=1..n} 1 + (1+x)^k ) / 3^(n+1).

%H Paul D. Hanna, <a href="/A338278/b338278.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: Sum_{n>=0} ( Product_{k=1..n} 1 + (1+x)^k ) / 3^(n+1).

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

%F a(n) = 3 (mod 9) for n >= 1 (conjecture).

%F a(n) ~ c * d^n * n! / sqrt(n), where d = 6.79252412537713528039794944605685... and c = 0.4829777246139702860411263158... - _Vaclav Kotesovec_, Oct 24 2020

%e G.f.: A(x) = 1 + 3*x + 30*x^2 + 507*x^3 + 12027*x^4 + 367167*x^5 + 13706049*x^6 + 604806546*x^7 + 30798675507*x^8 + 1777651129164*x^9 + 114681681511572*x^10 + ...

%e where

%e A(x) = 1/3 + (1 + (1+x))/3^2 + (1 + (1+x))*(1 + (1+x)^2)/3^3 + (1 + (1+x))*(1 + (1+x)^2)*(1 + (1+x)^3)/3^4 + (1 + (1+x))*(1 + (1+x)^2)*(1 + (1+x)^3)*(1 + (1+x)^4)/3^5 + ... + (Product_{k=1..n} 1 + (1+x)^k)/3^(n+1) + ...

%e also

%e A(x) = 1/2 + (1+x)/(2*(3 - (1+x))) + (1+x)^3/(2*(3 - (1+x))*(3 - (1+x)^2)) + (1+x)^6/(2*(3 - (1+x))*(3 - (1+x)^2)*(3 - (1+x)^3)) + (1+x)^10/(2*(3 - (1+x))*(3 - (1+x)^2)*(3 - (1+x)^3)*(3 - (1+x)^4)) + ... + (1+x)^(n*(n+1)/2)/(Product_{k=0..n} 3 - (1+x)^k) + ...

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 20 2020