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A098479
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Expansion of 1/sqrt((1-x)^2 - 4*x^3).
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10
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1, 1, 1, 3, 7, 13, 27, 61, 133, 287, 633, 1407, 3121, 6943, 15517, 34755, 77959, 175213, 394499, 889461, 2007963, 4538485, 10269247, 23258881, 52726599, 119627977, 271624315, 617180533, 1403272799, 3192557561, 7267485523, 16552454205
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OFFSET
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0,4
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COMMENTS
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1/sqrt((1-x)^2-4*r*x^3) expands to Sum_{k=0..floor(n/2)} binomial(n-k,k)*binomial(n-2*k,k)*r^k.
Apparently the number of lattice paths from (0,0) to (n,n) using steps (3,0), (0,3), (1,1).
It appears that 1/sqrt((1-x)^2-4*x^s) is the g.f. for lattice paths from (0,0) to (n,n) using steps (s,0), (0,s), (1,1).
Apparently the number of lattice paths from (0,0) to (n,n) using steps (1,2), (2,1), (1,1). (End)
Diagonal of rational functions 1/(1 - (x*y + x*y^2 + x^2*y)), 1/(1 - (x*y + x^3 + y^3)). - Gheorghe Coserea, Aug 31 2018
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n-2*k, k).
D-finite with recurrence: n*a(n) + (-2*n+1)*a(n-1) + (n-1)*a(n-2) + 2*(-2*n+3)*a(n-3) = 0. - R. J. Mathar, Nov 30 2012
G.f.: 1/(1 - x - 2*x^3/(1 - x - x^3/(1 - x - x^3/(1 - x - x^3/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021
a(n) ~ 1 / (sqrt((1-r)*(3-r)) * sqrt(Pi*n) * r^n), where r = 0.432040800333095... is the real root of the equation -1 + 2*r - r^2 + 4*r^3 = 0. - Vaclav Kotesovec, Jun 05 2022
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EXAMPLE
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The triangle of lattice paths from (0,0) to (n,k) using steps (1,2), (2,1), (1,1) begins
1;
0, 1;
0, 1, 1;
0, 0, 2, 3;
0, 0, 1, 3, 7;
0, 0, 0, 3, 7, 13;
0, 0, 0, 1, 6, 17, 27;
0, 0, 0, 0, 4, 14, 36, 61;
The triangle of lattice paths from (0,0) to (n,k) using steps (3,0), (0,3), (1,1) begins
1;
0, 1;
0, 0, 1;
1, 0, 0, 3;
0, 2, 0, 0, 7;
0, 0, 3, 0, 0, 13;
1, 0, 0, 7, 0, 0, 27;
0, 3, 0, 0, 17, 0, 0, 61;
The diagonals of both appear to be this sequence. (End)
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MATHEMATICA
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a[n_] := Sum[ Binomial[n-k, k]*Binomial[n-2k, k], {k, 0, n/2}]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jan 07 2013, from 1st formula *)
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PROG
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(PARI) /* as lattice paths, assuming the first comment is true */
/* same as in A092566 but use either of the following */
steps=[[3, 0], [0, 3], [1, 1]];
steps=[[1, 1], [1, 2], [2, 1]];
(Python)
from sympy import binomial
def a(n): return sum(binomial(n - k, k) * binomial(n - 2*k, k) for k in range(n//2 + 1))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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