A373928 Number of compositions of 7*n-2 into parts 1 and 7.
1, 7, 35, 168, 819, 4025, 19796, 97315, 478304, 2350860, 11554621, 56791883, 279136551, 1371977475, 6743373646, 33144194898, 162906243014, 800696596250, 3935484773527, 19343207491818, 95073338508548, 467292702057555, 2296779231936167, 11288844908179562
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).
Programs
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Mathematica
a[n_]:= n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*HypergeometricPFQ[{1-n, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6, (10+n)/6}, {6/7, 8/7, 9/7, 10/7, 11/7, 12/7}, -6^6/7^7]/120; Array[a,24] (* Stefano Spezia, Jun 23 2024 *) LinearRecurrence[{8,-21,35,-35,21,-7,1},{1,7,35,168,819,4025,19796},40] (* Harvey P. Dale, Jul 28 2024 *) -
PARI
a(n) = sum(k=0, n, binomial(n+4+6*k, n-1-k));
Formula
a(n) = A005709(7*n-2).
a(n) = Sum_{k=0..n} binomial(n+4+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)/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*hypergeom([1-n, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6, (10+n)/6], [6/7, 8/7, 9/7, 10/7, 11/7, 12/7], -6^6/7^7)/120. - Stefano Spezia, Jun 23 2024