A373932 Number of compositions of 7*n-6 into parts 1 and 7.
1, 3, 13, 66, 330, 1624, 7973, 39173, 192539, 946375, 4651541, 22862658, 112371609, 552314945, 2714670141, 13342810843, 65580931949, 322335276473, 1584302440665, 7786967198052, 38273537040452, 188117350476413, 924611109563490, 4544534046237850
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).
Programs
-
Mathematica
a[n_]:=n*HypergeometricPFQ[{1-n,(1+n)/6,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6}, {2/7, 3/7, 4/7, 5/7, 6/7, 8/7}, -6^6/7^7]; Array[a,24] (* Stefano Spezia, Jun 23 2024 *) -
PARI
a(n) = sum(k=0, n, binomial(n+6*k, n-1-k));
Formula
a(n) = A005709(7*n-6).
a(n) = Sum_{k=0..n} binomial(n+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)^5/((1-x)^7 - x).
a(n) = n*hypergeom([1-n,(1+n)/6,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6], [2/7, 3/7, 4/7, 5/7, 6/7, 8/7], -6^6/7^7). - Stefano Spezia, Jun 23 2024