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Number of compositions of 7*n into parts 6 and 7.
12

%I #14 Jun 22 2024 14:11:38

%S 1,1,1,1,1,1,2,9,37,121,331,793,1718,3448,6556,12121,22509,43453,

%T 89150,193823,436304,989759,2219064,4869285,10434412,21900170,

%U 45297211,93054446,191371581,396480142,830227401,1756883373,3746468095,8017653633,17151612398

%N Number of compositions of 7*n into parts 6 and 7.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-6,1).

%F a(n) = A017847(7*n).

%F a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-6*k).

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

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

%o (PARI) a(n) = sum(k=0, n\6, binomial(n+k, n-6*k));

%Y Cf. A373907, A373908, A373909, A373910, A373911.

%Y Cf. A099099, A099131, A107025, A373913.

%Y Cf. A017847.

%K nonn,easy

%O 0,7

%A _Seiichi Manyama_, Jun 22 2024