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Expansion of (1-5*x+6*x^2-x^3)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
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%I #17 Feb 20 2024 14:42:32

%S 1,4,14,48,165,572,2002,7071,25176,90251,325358,1178291,4282811,

%T 15612092,57040186,208772476,765186422,2807556411,10309833845,

%U 37883902913,139275229088,512223805060,1884404481767,6934058102453,25519786076294

%N Expansion of (1-5*x+6*x^2-x^3)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

%C In Pascal's triangle, subtract the 6th column to the left of the central column from the 2nd column.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-28,35,-15,1).

%F a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5); a(0) = 1, a(1) = 4, a(2) = 14, a(3) = 48, a(4) = 165.

%e a(0) = binomial(2,0);

%e a(1) = binomial(4,1);

%e a(2) = binomial(6,2) - binomial(6,0);

%e a(3) = binomial(8,3) - binomial(8,1);

%e a(4) = binomial(10,4) - binomial(10,2).

%t LinearRecurrence[{9, -28, 35, -15, 1}, {1, 4, 14, 48, 165}, 30] (* _Paolo Xausa_, Feb 20 2024 *)

%Y Cf. A007318, A211216, A224422, A221863, A122588.

%K nonn,easy

%O 0,2

%A _Peter Morris_, Feb 08 2024