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Number of compositions of 7*n-6 into parts 1 and 7.
4

%I #13 Jun 23 2024 10:32:30

%S 1,3,13,66,330,1624,7973,39173,192539,946375,4651541,22862658,

%T 112371609,552314945,2714670141,13342810843,65580931949,322335276473,

%U 1584302440665,7786967198052,38273537040452,188117350476413,924611109563490,4544534046237850

%N Number of compositions of 7*n-6 into parts 1 and 7.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,35,-35,21,-7,1).

%F a(n) = A005709(7*n-6).

%F a(n) = Sum_{k=0..n} binomial(n+6*k,n-1-k).

%F a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

%F G.f.: x*(1-x)^5/((1-x)^7 - x).

%F a(n) = n*hypergeom([1-n,(1+n)/6,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6], [2/7, 3/7, 4/7, 5/7, 6/7, 8/7], -6^6/7^7). - _Stefano Spezia_, Jun 23 2024

%t a[n_]:=n*HypergeometricPFQ[{1-n,(1+n)/6,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6}, {2/7, 3/7, 4/7, 5/7, 6/7, 8/7}, -6^6/7^7]; Array[a,24] (* _Stefano Spezia_, Jun 23 2024 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n+6*k, n-1-k));

%Y Cf. A099253, A373907, A373928, A373929, A373930, A373931.

%Y Cf. A005709.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Jun 23 2024