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Number of compositions of 5*n-3 into parts 4 and 5.
3

%I #17 Mar 15 2024 07:25:05

%S 0,0,1,4,10,20,36,65,129,285,661,1521,3394,7359,15706,33457,71760,

%T 155248,337745,735723,1600062,3471298,7517368,16269333,35221440,

%U 76302420,165392856,358594516,777451793,1685254125,3652428851,7915135727,17152718719,37172960128

%N Number of compositions of 5*n-3 into parts 4 and 5.

%H Paolo Xausa, <a href="/A369851/b369851.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-4,1).

%F a(n) = A017827(5*n-3).

%F a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,n-3-4*k).

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

%F G.f.: x^3*(1-x)/((1-x)^5 - x^4).

%t LinearRecurrence[{5, -10, 10, -4, 1}, {0, 0, 1, 4, 10}, 50] (* _Paolo Xausa_, Mar 15 2024 *)

%o (PARI) a(n) = sum(k=0, n\4, binomial(n+k, n-3-4*k));

%Y Cf. A099131, A368475, A369849, A369850.

%Y Cf. A017827.

%K nonn,easy

%O 1,4

%A _Seiichi Manyama_, Feb 03 2024