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A226405
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Expansion of x/((1-x-x^3)*(1-x)^3).
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7
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0, 1, 4, 10, 21, 40, 71, 120, 196, 312, 487, 749, 1139, 1717, 2571, 3830, 5683, 8407, 12408, 18281, 26898, 39537, 58071, 85245, 125082, 183478, 269074, 394534, 578418, 847927, 1242926, 1821840, 2670295, 3913782, 5736217, 8407142, 12321590, 18058510, 26466393
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OFFSET
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0,3
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COMMENTS
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A077868(n-1)= a(n) -2*a(n-1) +a(n-2), n>1.
A000217(n) = a(n) -a(n-1) -a(n-3), n>2.
A000930(n-1)= a(n) -3*a(n-1) +3*a(n-2) -a(n-3), n>2.
n = a(n) -2*a(n-1) +a(n-2) -a(n-3) +a(n-4), n>3.
1 = a(n) -3*a(n-1) +3*a(n-2) -2*a(n-3) +2*a(n-4) -a(n-5), n>4.
0 = a(n) -4*a(n-1) +6*a(n-2) -5*a(n-3) +4*a(n-4) -3*a(n-5) +a(n-6), n>5.
(End)
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LINKS
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FORMULA
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G.f.: x/((1-x-x^3)*(1-x)^3).
a(n) = Sum_{j=0..floor((n+2)/3)} binomial(n-2*j+2, j+3).
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MAPLE
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a:= n-> (Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -6, 5, -4, 3, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
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MATHEMATICA
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LinearRecurrence[{4, -6, 5, -4, 3, -1}, {0, 1, 4, 10, 21, 40}, 40] (* Bruno Berselli, Jun 07 2013 *)
CoefficientList[Series[x/((1-x-x^3)*(1-x)^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)
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PROG
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(PARI) my(x='x+O('x^50)); Vec(x/((1-x-x^3)*(1-x)^3)) \\ G. C. Greubel, Apr 28 2017
(Magma)
A226405:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+2, j+3): j in [0..Floor((n+2)/3)]]) >;
(SageMath)
def A226405(n): return sum(binomial(n-2*j+2, j+3) for j in (0..((n+2)//3)))
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CROSSREFS
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Cf. A000930, A050228, A077868, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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