A392269 - OEIS

%I #23 Jan 12 2026 17:53:34

%S 1,0,3,6,6,30,25,90,120,238,441,672,1333,2016,3681,5946,9945,16698,

%T 26876,45012,72216,118524,191178,308346,497559,795318,1276752,2033294,

%U 3242088,5150502,8166487,12934446,20427141,32240122,50764914,79850106,125402323,196667538

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

%H Seiichi Manyama, <a href="/A392269/b392269.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,3,-15,-12,17,18,-9,-11,3,3,-1).

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

%F a(n) = 6*a(n-2) + 3*a(n-3) - 15*a(n-4) - 12*a(n-5) + 17*a(n-6) + 18*a(n-7) - 9*a(n-8) - 11*a(n-9) + 3*a(n-10) + 3*a(n-11) - a(n-12).

%t CoefficientList[Series[(1-x^2)*((1-x^2)^2+3*x^3)/((1-x^2)^2-x^3)^3,{x,0,60}],x] (* _Vincenzo Librandi_, Jan 07 2026 *)

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

%o (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-x^2) * ((1-x^2)^2 + 3*x^3) / ((1-x^2)^2 - x^3)^3); // _Vincenzo Librandi_, Jan 07 2026

%Y Cf. A391963, A392252, A392270.

%Y Cf. A062200, A392267.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Jan 05 2026