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A226405 Expansion of x/((1-x-x^3)*(1-x)^3). 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
From Bruno Berselli, Jun 07 2013: (Start)
A050228(n) = a(n) -a(n-1), n>0.
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)
LINKS
FORMULA
G.f.: x/((1-x-x^3)*(1-x)^3).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+2)/3)} binomial(n-2*j+2, j+3).
a(n) = A099567(n+2, 3). (End)
MAPLE
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);
MATHEMATICA
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 *)
PROG
(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)]]) >;
[A226405(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022
(SageMath)
def A226405(n): return sum(binomial(n-2*j+2, j+3) for j in (0..((n+2)//3)))
[A226405(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022
CROSSREFS
4th column of A144903.
Sequence in context: A024988 A301174 A220907 * A144897 A001891 A266355
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Jun 06 2013
STATUS
approved

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Last modified May 23 22:02 EDT 2024. Contains 372765 sequences. (Running on oeis4.)