A373929 - OEIS

%I #17 Jun 23 2024 10:31:52

%S 1,6,28,133,651,3206,15771,77519,380989,1872556,9203761,45237262,

%T 222344668,1092840924,5371396171,26400821252,129762048116,

%U 637790353236,3134788177277,15407722718291,75730131016730,372219363549007,1829486529878612,8992065676243395

%N Number of compositions of 7*n-3 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-3).

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

%F a(n) = n*(1 + n)*(2 + n)*(3 + n)*hypergeom([1-n, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6], [5/7, 6/7, 8/7, 9/7, 10/7, 11/7], -6^6/7^7)/24. - _Stefano Spezia_, Jun 23 2024

%t a[n_]:=n*(1 + n)*(2 + n)*(3 + n)*HypergeometricPFQ[{1-n, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6}, {5/7, 6/7, 8/7, 9/7, 10/7, 11/7}, -6^6/7^7]/24; Array[a,24] (* _Stefano Spezia_, Jun 23 2024 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n+3+6*k, n-1-k));

%Y Cf. A099253, A373907, A373928, A373930, A373931, A373932.

%Y Cf. A005709, A369806.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Jun 23 2024