A098482 - OEIS

%I #28 Nov 19 2021 09:45:23

%S 1,1,1,1,3,7,13,21,37,73,147,283,531,1007,1953,3817,7423,14371,27877,

%T 54333,106189,207585,405743,793719,1554889,3049525,5984803,11751067,

%U 23086695,45388291,89289765,175746797,346077153,681795925,1343790319

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

%C From _Joerg Arndt_, Jul 01 2011: (Start)

%C Empirical: Number of lattice paths from (0,0) to (n,n) using steps (4,0), (0,4), (1,1).

%C 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).

%C Empirical: Number of lattice paths from (0,0) to (n,n) using steps (3,1), (1,3), (1,1).  (End)

%C 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 ).

%H Vincenzo Librandi, <a href="/A098482/b098482.txt">Table of n, a(n) for n = 0..200</a>

%H Steffen Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 2015.

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

%F 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

%F a(n) ~ 2^(n+1/2) / sqrt(3*Pi*n). - _Vaclav Kotesovec_, Jun 23 2014

%F 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

%e From _Joerg Arndt_, Jul 01 2011: (Start)

%e The triangle of lattice paths from (0,0) to (n,k) using steps (3,1), (1,3), (1,1) begins

%e 1;

%e 0, 1;

%e 0, 0, 1;

%e 0, 1, 0, 1;

%e 0, 0, 2, 0, 3;

%e 0, 0, 0, 3, 0, 7;

%e 0, 0, 1, 0, 4, 0, 13;

%e 0, 0, 0, 3, 0, 8, 0, 21;

%e 0, 0, 0, 0, 6, 0, 18, 0, 37;

%e 0, 0, 0, 1, 0, 10, 0, 37, 0, 73;

%e The triangle of lattice paths from (0,0) to (n,k) using steps (4,0), (0,4), (1,1) begins

%e 1;

%e 0, 1;

%e 0, 0, 1;

%e 0, 0, 0, 1;

%e 1, 0, 0, 0, 3;

%e 0, 2, 0, 0, 0, 7;

%e 0, 0, 3, 0, 0, 0, 13;

%e 0, 0, 0, 4, 0, 0, 0, 21;

%e 1, 0, 0, 0, 8, 0, 0, 0, 37;

%e 0, 3, 0, 0, 0, 18, 0, 0, 0, 73;

%e The diagonals of both appear to be this sequence.  (End)

%p 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

%t CoefficientList[Series[1/Sqrt[(1-x)^2-4*x^4], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 23 2014 *)

%o (PARI) /* as lattice paths, assuming the first comment is true */

%o /* same as in A092566 but use either of */

%o steps=[[4,0], [0,4], [1,1]];

%o steps=[[3,1], [1,3], [1,1]];

%o /* Joerg Arndt, Jul 01 2011 */

%Y Cf. A098479, A098483, A098484.

%K easy,nonn

%O 0,5

%A _Paul Barry_, Sep 10 2004