A190153 Row sums of the triangle A190152.
1, 2, 12, 65, 351, 1897, 10252, 55405, 299426, 1618192, 8745217, 47261895, 255418101, 1380359512, 7459895657, 40315615410, 217878227876, 1177482265857, 6363483400447, 34390259761825, 185855747875876, 1004422742303477, 5428215467030962
Offset: 0
Links
- Nathaniel Johnston, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (5,2,1).
Programs
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Magma
I:=[1, 2, 12]; [n le 3 select I[n] else 5*Self(n-1) +2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 30 2017
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Maple
seq(add(binomial(3*n-k,3*n-3*k), k=0..n), n=0..20);
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Mathematica
Table[Sum[Binomial[3n - k, 3n - 3k], {k, 0, n}], {n, 0, 22}] LinearRecurrence[{5,2,1}, {1,2,12}, 30] (* G. C. Greubel, Dec 30 2017 *) -
Maxima
makelist(sum(binomial(3*n-k,3*n-3*k),k,0,n),n,0,22);
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PARI
x='x+O('x^30); Vec((1-3*x)/(1-5*x-2*x^2-x^3)) \\ G. C. Greubel, Dec 30 2017
Formula
a(n) = Sum_{k=0..n} binomial(3*n-k,3*n-3*k).
From Colin Barker, Mar 21 2012: (Start)
a(n) = 5*a(n-1) + 2*a(n-2) + a(n-3).
G.f.: (1-3*x)/(1-5*x-2*x^2-x^3). (End)