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Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^3) ).
1

%I #10 Aug 13 2024 11:41:51

%S 1,-2,4,-3,6,-2,14,3,32,35,92,142,309,541,1061,1970,3770,7067,13423,

%T 25328,47925,90546,171268,323704,612034,1157045,2187523,4135499,

%U 7818493,14781207,27944635,52830674,99879267,188826659,356986436,674901081,1275934925,2412219595

%N Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^3) ).

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,5,10,10,5,1).

%F a(n) = -2*a(n-1) + 5*a(n-3) + 10*a(n-4) + 10*a(n-5) + 5*a(n-6) + a(n-7).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(3*k-2,n-2*k).

%o (PARI) my(N=40, x='x+O('x^N)); Vec(1/((1+x)^2*(1-x^2*(1+x)^3)))

%o (PARI) a(n) = sum(k=0, n\2, binomial(3*k-2, n-2*k));

%Y Cf. A375315, A375317, A375364.

%K sign

%O 0,2

%A _Seiichi Manyama_, Aug 13 2024