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Number of partitions of 3^n into parts that are at most 3.
2

%I #23 Jun 26 2024 04:04:07

%S 1,3,12,75,588,5043,44652,399675,3590508,32294883,290594892,

%T 2615176875,23536060428,211822949523,1906401762732,17157601515675,

%U 154418370594348,1389765206208963,12507886468460172,112570977053880075,1013138789998136268,9118249099522873203

%N Number of partitions of 3^n into parts that are at most 3.

%H Alois P. Heinz, <a href="/A238630/b238630.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (13,-39,27).

%F a(n) = [x^(3^n)] Product_{j=1..3} 1/(1-x^j).

%F G.f.: (9*x^3+12*x^2-10*x+1)/((1-x)*(1-3*x)*(1-3^2*x)).

%F a(n) = A001399(3^n) = round((3^n+3)^2/12).

%F a(n) = 3*A051500(n-1) for n>=1. - _Hugo Pfoertner_, May 04 2024

%e a(2) = 12: 333, 3222, 3321, 22221, 32211, 33111, 222111, 321111, 2211111, 3111111, 21111111, 111111111.

%p gf:= (9*x^3+12*x^2-10*x+1)/((1-x)*(1-3*x)*(1-3^2*x)):

%p a:= n-> coeff(series(gf, x, n+1), x, n):

%p seq(a(n), n=0..30);

%t Round[(3^Range[0, 25] + 3)^2/12] (* _Paolo Xausa_, Jun 26 2024 *)

%Y Row n=3 of A238016.

%Y Cf. A001399, A051500.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Mar 01 2014