A373929 Number of compositions of 7*n-3 into parts 1 and 7.

1, 6, 28, 133, 651, 3206, 15771, 77519, 380989, 1872556, 9203761, 45237262, 222344668, 1092840924, 5371396171, 26400821252, 129762048116, 637790353236, 3134788177277, 15407722718291, 75730131016730, 372219363549007, 1829486529878612, 8992065676243395

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-3).

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

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Cf. A005709, A369806.

Sequence in context: A287807 A155588 A368574 * A208439 A108051 A199315

Adjacent sequences: A373926 A373927 A373928 * A373930 A373931 A373932

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 23 2024

Status

approved