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Expansion of 1/(1 - x^2/(1-x)^7).
5

%I #16 Feb 02 2024 16:14:27

%S 1,0,1,7,29,98,316,1043,3536,12083,41168,139750,473824,1607014,

%T 5453022,18506947,62808496,213144034,723295969,2454483506,8329290739,

%U 28265565587,95919580313,325504019213,1104600373788,3748469764612,12720462563684,43166996581876

%N Expansion of 1/(1 - x^2/(1-x)^7).

%C Number of compositions of 7*n-2 into parts 2 and 7.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-20,35,-35,21,-7,1).

%F a(n) = A369813(7*n-2) for n > 0.

%F a(n) = 7*a(n-1) - 20*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+5*k,n-2*k).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-x)^7))

%o (PARI) a(n) = sum(k=0, n\2, binomial(n-1+5*k, n-2*k));

%Y Cf. A099253, A369806, A369807, A369808, A369809.

%Y Cf. A369813.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Feb 01 2024