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A373930
Number of compositions of 7*n-4 into parts 1 and 7.
4
1, 5, 22, 105, 518, 2555, 12565, 61748, 303470, 1491567, 7331205, 36033501, 177107406, 870496256, 4278555247, 21029425081, 103361226864, 508028305120, 2496997824041, 12272934541014, 60322408298439, 296489232532277, 1457267166329605, 7162579146364783
OFFSET
1,2
FORMULA
a(n) = A005709(7*n-4).
a(n) = Sum_{k=0..n} binomial(n+2+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)^3/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*hypergeom([1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6], [4/7, 5/7, 6/7, 8/7, 9/7, 10/7], -6^6/7^7)/6. - Stefano Spezia, Jun 23 2024
MATHEMATICA
a[n_]:=n*(1 + n)*(2 + n)*HypergeometricPFQ[{1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6}, {4/7, 5/7, 6/7, 8/7, 9/7, 10/7}, -6^6/7^7]/6; Array[a, 24] (* Stefano Spezia, Jun 23 2024 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+2+6*k, n-1-k));
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Jun 23 2024
STATUS
approved