A373931 - OEIS

%I #16 Jun 23 2024 10:32:09

%S 1,4,17,83,413,2037,10010,49183,241722,1188097,5839638,28702296,

%T 141073905,693388850,3408058991,16750869834,82331801783,404667078256,

%U 1988969518921,9775936716973,48049473757425,236166824233838,1160777933797328,5705311980035178

%N Number of compositions of 7*n-5 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-5).

%F a(n) = Sum_{k=0..n} binomial(n+1+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)^4/((1-x)^7 - x).

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

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

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

%Y Cf. A099253, A373907, A373928, A373929, A373930, A373932.

%Y Cf. A005709, A369808.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Jun 23 2024