A392267 - OEIS

%I #19 Jan 16 2026 10:00:26

%S 1,0,2,3,3,12,9,30,35,67,111,161,296,420,728,1103,1764,2802,4313,6897,

%T 10588,16691,25830,40137,62365,96225,149235,229840,354936,546400,

%U 840653,1292760,1984226,3045967,4668159,7152380,10947997,16746318,25601471,39107943

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

%H Seiichi Manyama, <a href="/A392267/b392267.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,2,-6,-4,3,2,-1).

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

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

%t CoefficientList[Series[((1-x^2)^2+x^3)/((1-x^2)^2-x^3)^2,{x,0,60}],x] (* _Vincenzo Librandi_, Jan 16 2026 *)

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

%o (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R! ((1-x^2)^2 + x^3) / ((1-x^2)^2 - x^3)^2); // _Vincenzo Librandi_, Jan 16 2026

%Y Cf. A391962, A392251, A392268.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 05 2026