A373930 - OEIS

%I #15 Jun 23 2024 10:31:57

%S 1,5,22,105,518,2555,12565,61748,303470,1491567,7331205,36033501,

%T 177107406,870496256,4278555247,21029425081,103361226864,508028305120,

%U 2496997824041,12272934541014,60322408298439,296489232532277,1457267166329605,7162579146364783

%N Number of compositions of 7*n-4 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-4).

%F a(n) = Sum_{k=0..n} binomial(n+2+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)^3/((1-x)^7 - x).

%F a(n) = n*(1 + n)*(2 + n)*hypergeom([1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6], [4/7, 5/7, 6/7, 8/7, 9/7, 10/7], -6^6/7^7)/6. - _Stefano Spezia_, Jun 23 2024

%t a[n_]:=n*(1 + n)*(2 + n)*HypergeometricPFQ[{1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6}, {4/7, 5/7, 6/7, 8/7, 9/7, 10/7}, -6^6/7^7]/6; Array[a,24] (* _Stefano Spezia_, Jun 23 2024 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n+2+6*k, n-1-k));

%Y Cf. A099253, A373907, A373928, A373929, A373931, A373932.

%Y Cf. A005709, A369807.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Jun 23 2024