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A098479
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Expansion of 1/sqrt((1-x)^2-4*x^3).
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7
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| From Joerg Arndt, Jul 01 2011: (Start)
Empirical: 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).
Empirical: Number of lattice paths from (0,0) to (n,n) using steps (1,2), (2,1), (1,1). (End)
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}
Hankel transform is A120580. [From Paul Barry (pbarry(AT)wit.ie), Sep 19 2008]
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REFERENCES
| J. Cigler, Some nice Hankel determinants. Arxiv preprint arXiv:1109.1449, 2011.
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FORMULA
| a(n) = sum(k=0..floor(n/2), binomial(n-k, k)*binomial(n-2*k, k) ).
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EXAMPLE
| From Joerg Arndt, Jul 01 2011: (Start)
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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PROG
| (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]];
/* Joerg Arndt, Jul 01 2011 */
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CROSSREFS
| Cf. A098480, A098481.
Sequence in context: A068673 A140465 A080241 * A119445 A146904 A146432
Adjacent sequences: A098476 A098477 A098478 * A098480 A098481 A098482
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 10 2004
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