A338278 G.f.: Sum_{n>=0} ( Product_{k=1..n} 1 + (1+x)^k ) / 3^(n+1).

1, 3, 30, 507, 12027, 367167, 13706049, 604806546, 30798675507, 1777651129164, 114681681511572, 8177632124654658, 638681988964923723, 54220733932072259388, 4971400927192150334778, 489591085640900144694105, 51541568078546978411976714, 5776152891297758939283199518

Offset

0,2

Links

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

Formula

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

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

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

a(n) ~ c * d^n * n! / sqrt(n), where d = 6.79252412537713528039794944605685... and c = 0.4829777246139702860411263158... - Vaclav Kotesovec, Oct 24 2020

Example

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 + ...
where
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) + ...
also
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) + ...

Crossrefs

Sequence in context: A201466 A064352 A366002 * A144739 A185827 A366009

Adjacent sequences: A338275 A338276 A338277 * A338279 A338280 A338281

Keyword

nonn

Author

Paul D. Hanna, Oct 20 2020

Status

approved