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A227430
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Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).
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1
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0, 0, 1, 3, 6, 10, 15, 21, 29, 45, 90, 220, 561, 1365, 3095, 6555, 13110, 25126, 46971, 87381, 164921, 320001, 640002, 1309528, 2707629, 5592405, 11450531, 23166783, 46333566, 91869970, 181348455, 357913941, 708653429, 1410132405, 2820264810, 5662052980
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OFFSET
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0,4
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COMMENTS
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Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences:
0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n): after 0, it is A192080.
0, 0, 0, 0, 1, 5, 15, 35, 70, 126,... e(n)
0, 0, 0, 1, 4, 10, 20, 35, 56, 85,... f(n)
0, 0, 1, 3, 6, 10, 15, 21, 29, 45,... a(n)
0, 1, 2, 3, 4, 5, 6, 8, 16, 45,... b(n)
1, 1, 1, 1, 1, 1, 2, 8, 29, 85,... c(n)
0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n).
a(n) - d(n) = 0, 0, 1, 3, 6, 9, 9, 0,... A057083(n-2)
b(n) - e(n) = 0, 1, 2, 3, 3, 0, -9, -27,... A057682(n)
c(n) - f(n) = 1, 1, 1, 0, -3, -9, -18, -27,... A057681(n)
d(n) - a(n) = 0, 0, -1, -3, -6, -9, -9, 0,... -A057083(n-2)
e(n) - b(n) = 0, -1, -2, -3, -3, 0, 9, 27,... -A057682(n)
f(n) - c(n) = -1, -1, -1, 0, 3, 9, 18, 27,... -A057681(n).
The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9 A029898(n)?
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LINKS
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FORMULA
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a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10.
G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - Ralf Stephan, Jul 13 2013
a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - Seiichi Manyama, Mar 23 2019
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EXAMPLE
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a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.
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MATHEMATICA
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Join[{0}, LinearRecurrence[{6, -15, 20, -15, 6}, {0, 1, 3, 6, 10}, 40]] (* Harvey P. Dale, Dec 17 2014 *)
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PROG
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(PARI) {a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ Seiichi Manyama, Mar 23 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Definition uses the g.f. of Ralf Stephan.
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STATUS
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approved
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