A098482 Expansion of 1/sqrt((1-x)^2-4*x^4).

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

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 ).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.

Formula

a(n) = sum(k=0..floor(n/2), binomial(n-2*k, k)*binomial(n-3*k, k) ).

D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 4*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014

a(n) ~ 2^(n+1/2) / sqrt(3*Pi*n). - Vaclav Kotesovec, Jun 23 2014

G.f.: 1/(1 - x - 2*x^4/(1 - x - x^4/(1 - x - x^4/(1 - x - x^4/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021

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, Apr 03 2007

Mathematica

CoefficientList[Series[1/Sqrt[(1-x)^2-4*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 23 2014 *)

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: A138035 A355736 A032606 * A342422 A147432 A018367

Adjacent sequences: A098479 A098480 A098481 * A098483 A098484 A098485

Keyword

easy,nonn

Author

Paul Barry, Sep 10 2004

Status

approved