|
| |
|
|
A098482
|
|
Expansion of 1/sqrt((1-x)^2-4*x^4).
|
|
3
| |
|
|
1, 1, 1, 1, 3, 7, 13, 21, 37, 73, 147, 283, 531, 1007, 1953, 3817, 7423, 14371, 27877, 54333, 106189, 207585, 405743, 793719, 1554889, 3049525, 5984803, 11751067, 23086695, 45388291, 89289765, 175746797, 346077153, 681795925, 1343790319
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,5
|
|
|
COMMENTS
| From Joerg Arndt, Jul 01 2011: (Start)
Empirical: Number of lattice paths from (0,0) to (n,n) using steps (4,0), (0,4), (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 (3,1), (1,3), (1,1). (End)
1/sqrt((1-x)^2-4*r*x^4) expands to sum(k=0..floor(n/2), binomial(n-2*k,k)*binomial(n-3*k,k)*r^k ).
|
|
|
FORMULA
| a(n) = sum(k=0..floor(n/2), binomial(n-2*k, k)*binomial(n-3*k, k) ).
|
|
|
EXAMPLE
| From Joerg Arndt, Jul 01 2011: (Start)
The triangle of lattice paths from (0,0) to (n,k) using steps (3,1), (1,3), (1,1) begins
1;
0, 1;
0, 0, 1;
0, 1, 0, 1;
0, 0, 2, 0, 3;
0, 0, 0, 3, 0, 7;
0, 0, 1, 0, 4, 0, 13;
0, 0, 0, 3, 0, 8, 0, 21;
0, 0, 0, 0, 6, 0, 18, 0, 37;
0, 0, 0, 1, 0, 10, 0, 37, 0, 73;
The triangle of lattice paths from (0,0) to (n,k) using steps (4,0), (0,4), (1,1) begins
1;
0, 1;
0, 0, 1;
0, 0, 0, 1;
1, 0, 0, 0, 3;
0, 2, 0, 0, 0, 7;
0, 0, 3, 0, 0, 0, 13;
0, 0, 0, 4, 0, 0, 0, 21;
1, 0, 0, 0, 8, 0, 0, 0, 37;
0, 3, 0, 0, 0, 18, 0, 0, 0, 73;
The diagonals of both appear to be this sequence. (End)
|
|
|
MAPLE
| seq(add(binomial(n-3*k, k)*binomial(n-2*k, k), k=0..floor(n/3)), n=0..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2007
|
|
|
PROG
| (PARI) /* as lattice paths, assuming the first comment is true */
/* same as in A092566 but use either of */
steps=[[4, 0], [0, 4], [1, 1]];
steps=[[3, 1], [1, 3], [1, 1]];
/* Joerg Arndt, Jul 01 2011 */
|
|
|
CROSSREFS
| Cf. A098479, A098483, A098484.
Sequence in context: A098575 A138035 A032606 * A147432 A018367 A146369
Adjacent sequences: A098479 A098480 A098481 * A098483 A098484 A098485
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 10 2004
|
| |
|
|
