A373931 Number of compositions of 7*n-5 into parts 1 and 7.

1, 4, 17, 83, 413, 2037, 10010, 49183, 241722, 1188097, 5839638, 28702296, 141073905, 693388850, 3408058991, 16750869834, 82331801783, 404667078256, 1988969518921, 9775936716973, 48049473757425, 236166824233838, 1160777933797328, 5705311980035178

Offset

1,2

Links

Table of n, a(n) for n=1..24.

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Formula

a(n) = A005709(7*n-5).

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

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).

G.f.: x*(1-x)^4/((1-x)^7 - x).

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Cf. A005709, A369808.

Sequence in context: A110771 A082028 A377960 * A052315 A369476 A200716

Adjacent sequences: A373928 A373929 A373930 * A373932 A373933 A373934

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 23 2024

Status

approved