%I #18 May 31 2021 13:11:21

%S 1,6,21,56,126,252,461,780,1209,1638,1638,0,-6187,-23238,-63783,

%T -151316,-326382,-652764,-1217483,-2107560,-3322995,-4538430,-4538430,

%U 0,16942381,63239286,172791861,408855776,880983606,1761967212,3287837741,5694626340

%N Expansion of 1/((1 - x)^6 + x^6).

%H Seiichi Manyama, <a href="/A306940/b306940.txt">Table of n, a(n) for n = 0..3000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-2).

%F a(n) = Sum_{k=0..floor(n/6)} (-1)^k*binomial(n+5,6*k+5).

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - 2*a(n-6) for n > 5.

%t CoefficientList[Series[1/((1 - x)^6 + x^6), {x, 0, 31}], x] (* _Amiram Eldar_, May 25 2021 *)

%t LinearRecurrence[{6,-15,20,-15,6,-2},{1,6,21,56,126,252},40] (* _Harvey P. Dale_, May 31 2021 *)

%o (PARI) {a(n) = sum(k=0, n\6, (-1)^k*binomial(n+5, 6*k+5))}

%o (PARI) N=66; x='x+O('x^N); Vec(1/((1-x)^6+x^6))

%Y Column 6 of A306914.

%K sign,easy

%O 0,2

%A _Seiichi Manyama_, Mar 17 2019