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Expansion of 1/((1 - x)^9 - x^9).
9

%I #19 May 26 2021 00:55:26

%S 1,9,45,165,495,1287,3003,6435,12870,24311,43776,75753,127110,209475,

%T 346104,591261,1081575,2163150,4686826,10656387,24582663,56191734,

%U 125640180,273241161,577147212,1184959314,2369918628,4631710931,8881943832,16798969548,31537530456

%N Expansion of 1/((1 - x)^9 - x^9).

%H Seiichi Manyama, <a href="/A306939/b306939.txt">Table of n, a(n) for n = 0..3000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,2).

%F a(n) = Sum_{k=0..floor(n/9)} binomial(n+8,9*k+8).

%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + 2*a(n-9) for n > 8.

%t CoefficientList[Series[1/((1 - x)^9 - x^9), {x, 0, 30}], x] (* _Amiram Eldar_, May 25 2021 *)

%o (PARI) {a(n) = sum(k=0, n\9, binomial(n+8, 9*k+8))}

%o (PARI) N=66; x='x+O('x^N); Vec(1/((1-x)^9-x^9))

%Y Column 9 of A306915.

%Y Cf. A306860.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Mar 17 2019