A293169 a(n) = Sum_{k=0..n} binomial(k, 6*(n-k)).
1, 1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 463, 925, 1718, 3017, 5097, 8464, 14197, 24753, 45697, 89150, 180254, 368734, 748924, 1493990, 2914906, 5565127, 10434412, 19322901, 35583926, 65615746, 121847272, 228638698, 433747259, 830227401, 1597653852, 3078928619, 5922703731, 11347651254
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
- Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1,1).
Programs
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Maple
f:=n-> add( binomial(k, 6*(n-k)), k=0..n); [seq(f(n),n=0..30)];
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Mathematica
Table[Sum[Binomial[k,6(n-k)],{k,0,n}],{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1,1},{1,1,1,1,1,1,1},50] (* Harvey P. Dale, Apr 10 2022 *) -
PARI
Vec((1 - x)^5 / (1 - 6*x + 15*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6 - x^7) + O(x^30)) \\ Colin Barker, Oct 18 2017
Formula
From Colin Barker, Oct 17 2017: (Start)
G.f.: (1 - x)^5 / (1 - 6*x + 15*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6 - x^7).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + a(n-7) for n>6.
(End)