A390221 - OEIS

%I #19 Jan 20 2026 14:16:06

%S 1,0,2,0,3,0,4,0,5,1,6,6,7,21,8,56,9,126,11,252,21,462,67,792,233,

%T 1287,729,2003,2017,3017,5021,4473,11457,6748,24328,10948,48640,20196,

%U 92416,42636,168152,97869,295092,229824,503428,531070,843548,1186592,1408476

%N a(n) = Sum_{k=0..floor(n/2)} binomial(k+1,4*n-8*k+1).

%H Seiichi Manyama, <a href="/A390221/b390221.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 (0,4,0,-6,0,4,0,-1,1).

%F G.f.: (1-x^2)^2 / ((1-x^2)^4 - x^9).

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

%F a(n) = A390222(n) - A390222(n-2).

%t CoefficientList[Series[(1-x^2)^2/((1-x^2)^4-x^9),{x,0,50}],x] (* _Vincenzo Librandi_, Jan 20 2026 *)

%o (PARI) my(N=50, x='x+O('x^N)); Vec((1-x^2)^2/((1-x^2)^4-x^9))

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R! (1-x^2)^2 / ((1-x^2)^4 - x^9)); // _Vincenzo Librandi_, Jan 20 2026

%Y Cf. A390040, A390220, A390222.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Jan 19 2026