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A227430
Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).
1
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
OFFSET
0,4
COMMENTS
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) = A024495(n),
b(n) + e(n) = A131708(n),
c(n) + f(n) = A024493(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 column is A131531(n).
The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9 A029898(n)?
FORMULA
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.
a(n) = A024495(n) - A192080(n-5) for n>4.
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
EXAMPLE
a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.
MATHEMATICA
Join[{0}, LinearRecurrence[{6, -15, 20, -15, 6}, {0, 1, 3, 6, 10}, 40]] (* Harvey P. Dale, Dec 17 2014 *)
PROG
(PARI) {a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ Seiichi Manyama, Mar 23 2019
CROSSREFS
Sequence in context: A175724 A335184 A101551 * A051166 A373934 A188278
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Jul 11 2013
EXTENSIONS
Definition uses the g.f. of Ralf Stephan.
More terms from Harvey P. Dale, Dec 17 2014
STATUS
approved