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

%I A098482 #42 Jul 11 2025 14:27:48
%S A098482 1,1,1,1,3,7,13,21,37,73,147,283,531,1007,1953,3817,7423,14371,27877,
%T A098482 54333,106189,207585,405743,793719,1554889,3049525,5984803,11751067,
%U A098482 23086695,45388291,89289765,175746797,346077153,681795925,1343790319,2649687079,5226711507
%N A098482 Expansion of 1/sqrt((1-x)^2 - 4*x^4).
%C A098482 From _Joerg Arndt_, Jul 01 2011: (Start)
%C A098482 Empirical: Number of lattice paths from (0,0) to (n,n) using steps (4,0), (0,4), (1,1).
%C A098482 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 A098482 Empirical: Number of lattice paths from (0,0) to (n,n) using steps (3,1), (1,3), (1,1).  (End)
%C A098482 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 ).
%C A098482 Diagonal of the rational function 1 / ((1-x)*(1-y) - x^3*y^4). - _Seiichi Manyama_, Apr 29 2025
%C A098482 a(n) is the number of ways to tile a strip of length n with 1 X 1 squares and 1 X 2 red dominos and 1 X 2 blue dominos, with an equal number of red and blue dominos. - _Greg Dresden_ and Leo Zhang, Jul 08 2025
%H A098482 Vincenzo Librandi, <a href="/A098482/b098482.txt">Table of n, a(n) for n = 0..200</a>
%H A098482 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 A098482 a(n) = Sum_{k=0..floor(n/4)} binomial(n-2*k, k) * binomial(n-3*k, k).
%F A098482 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 A098482 a(n) ~ 2^(n+1/2) / sqrt(3*Pi*n). - _Vaclav Kotesovec_, Jun 23 2014
%F A098482 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
%F A098482 a(n) = Sum_{k=0..floor(n/4)} binomial(n-2*k, 2*k) * binomial(2*k, k). - _Greg Dresden_ and Leo Zhang, Jul 08 2025
%e A098482 From _Joerg Arndt_, Jul 01 2011: (Start)
%e A098482 The triangle of lattice paths from (0,0) to (n,k) using steps (3,1), (1,3), (1,1) begins
%e A098482   1;
%e A098482   0, 1;
%e A098482   0, 0, 1;
%e A098482   0, 1, 0, 1;
%e A098482   0, 0, 2, 0, 3;
%e A098482   0, 0, 0, 3, 0, 7;
%e A098482   0, 0, 1, 0, 4, 0, 13;
%e A098482   0, 0, 0, 3, 0, 8, 0, 21;
%e A098482   0, 0, 0, 0, 6, 0, 18, 0, 37;
%e A098482   0, 0, 0, 1, 0, 10, 0, 37, 0, 73;
%e A098482 The triangle of lattice paths from (0,0) to (n,k) using steps (4,0), (0,4), (1,1) begins
%e A098482   1;
%e A098482   0, 1;
%e A098482   0, 0, 1;
%e A098482   0, 0, 0, 1;
%e A098482   1, 0, 0, 0, 3;
%e A098482   0, 2, 0, 0, 0, 7;
%e A098482   0, 0, 3, 0, 0, 0, 13;
%e A098482   0, 0, 0, 4, 0, 0, 0, 21;
%e A098482   1, 0, 0, 0, 8, 0, 0, 0, 37;
%e A098482   0, 3, 0, 0, 0, 18, 0, 0, 0, 73;
%e A098482 The diagonals of both appear to be this sequence.  (End)
%p A098482 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 A098482 CoefficientList[Series[1/Sqrt[(1-x)^2-4*x^4], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 23 2014 *)
%o A098482 (PARI) /* as lattice paths, assuming the first comment is true */
%o A098482 /* same as in A092566 but use either of */
%o A098482 steps=[[4,0], [0,4], [1,1]];
%o A098482 steps=[[3,1], [1,3], [1,1]];
%o A098482 /* Joerg Arndt, Jul 01 2011 */
%Y A098482 Cf. A098479, A098483, A098484.
%K A098482 easy,nonn
%O A098482 0,5
%A A098482 _Paul Barry_, Sep 10 2004