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Number of compositions of 6*n-5 into parts 1 and 6.
3

%I #11 Jun 24 2024 08:46:18

%S 1,3,12,55,251,1133,5103,22990,103598,466852,2103796,9480387,42721676,

%T 192517665,867546829,3909446467,17617229520,79388930909,357752184782,

%U 1612146986543,7264855441424,32737786954481,147527050375071,664804576516400,2995824317191471

%N Number of compositions of 6*n-5 into parts 1 and 6.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,20,-15,6,-1).

%F a(n) = A005708(6*n-5).

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

%F a(n) = 7*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

%F G.f.: x*(1-x)^4/((1-x)^6 - x).

%F a(n) = A373959(n) - A373959(n-1).

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

%Y Cf. A099242, A371125, A373302, A373958, A373959.

%Y Cf. A005708.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Jun 23 2024