A391903 - OEIS

%I #18 Dec 31 2025 04:14:46

%S 1,0,1,1,1,20,2,84,85,221,925,675,5007,5821,19020,49951,72829,296770,

%T 428528,1364693,3001497,5998390,18112232,31698490,92287620,193987770,

%U 443261263,1145617890,2292692419,6134650552,13029745624,31155756788,74677089095,162011895318,407612202760

%N a(n) = Sum_{k=0..floor(n/2)} binomial(3*k,3*(n-2*k)).

%H Seiichi Manyama, <a href="/A391903/b391903.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,3,-3,21,-2,3,3,1).

%F G.f.: ((1-x^2-x^3)^2 - 9*x^5) / ((1-x^2-x^3)^3 - 27*x^5).

%F a(n) = 3*a(n-2) + 3*a(n-3) - 3*a(n-4) + 21*a(n-5) - 2*a(n-6) + 3*a(n-7) + 3*a(n-8) + a(n-9).

%t CoefficientList[Series[((1-x^2-x^3)^2-9*x^5)/((1-x^2-x^3)^3-27*x^5),{x,0,50}],x] (* _Vincenzo Librandi_, Dec 30 2025 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(((1-x^2-x^3)^2-9*x^5)/((1-x^2-x^3)^3-27*x^5))

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^2-x^3)^2 - 9*x^5) / ((1-x^2-x^3)^3 - 27*x^5)); // _Vincenzo Librandi_, Dec 30 2025

%Y Cf. A391902, A391904.

%K nonn

%O 0,6

%A _Seiichi Manyama_, Dec 23 2025