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Expansion of 1/((1 - x)^9 + x^9).
2

%I #16 Sep 25 2019 16:42:33

%S 1,9,45,165,495,1287,3003,6435,12870,24309,43740,75411,124830,197505,

%T 293436,389367,389367,0,-1562274,-6216183,-18365697,-47600136,

%U -113879520,-257123889,-554298228,-1148646906,-2297293812,-4443424371,-8313049440,-15011769204

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

%H Seiichi Manyama, <a href="/A306942/b306942.txt">Table of n, a(n) for n = 0..3000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9).

%F a(n) = Sum_{k=0..floor(n/9)} (-1)^k*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) for n > 7.

%t CoefficientList[Series[1/((1-x)^9+x^9),{x,0,30}],x] (* _Harvey P. Dale_, Sep 25 2019 *)

%o (PARI) {a(n) = sum(k=0, n\9, (-1)^k*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 A306914.

%Y Cf. A306939.

%K sign,easy

%O 0,2

%A _Seiichi Manyama_, Mar 17 2019