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A122868 Expansion of 1/sqrt(1-6x-3x^2).

%I A122868 #61 Nov 13 2022 13:33:28
%S A122868 1,3,15,81,459,2673,15849,95175,576963,3523257,21640365,133549155,
%T A122868 827418645,5143397535,32063180535,200367960201,1254816463923,
%U A122868 7873205412825,49482344889261,311457546052659,1963051327342449,12387750763156227,78258731003169435
%N A122868 Expansion of 1/sqrt(1-6x-3x^2).
%C A122868 Binomial transform of A084609. Central coefficients of (1+3x+3x^2)^n.
%C A122868 The number of free (3,3)-Motzkin paths of length n, where free (k,t)-Motzkin paths are the free Motzkin paths with level steps of weight k and down steps of weight t. For example a(2)=15 because there are 9, 3, 3 paths consisting of two level steps, UD's and DU's, respectively. - Carol J. Wang (cerlined7(AT)hotmail.com), Nov 27 2007
%C A122868 The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - _Peter Bala_, Jan 07 2022
%H A122868 Alois P. Heinz, <a href="/A122868/b122868.txt">Table of n, a(n) for n = 0..1236</a> (first 201 terms from Vincenzo Librandi)
%H A122868 Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%H A122868 W. Y. C. Chen, N. Y. Li, L. W. Shapiro and S. H. F. Yan, <a href="http://dx.doi.org/10.1016/j.ejc.2006.02.005"> Matrix identities on weighted partial Motzkin paths</a>, European J. Combinatorics, 28 (2007), 1196-2007.
%H A122868 M. Dziemianczuk, <a href="http://dx.doi.org/10.1007/s00373-013-1357-1">Counting Lattice Paths With Four Types of Steps</a>, Graphs and Combinatorics, September 2013, Volume 30, Issue 6, pp 1427-1452.
%H A122868 M. Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
%H A122868 J. L. Ramírez and V. F. Sirvent, <a href="https://doi.org/10.37236/4618">A Generalization of the k-Bonacci Sequence from Riordan Arrays</a>, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.
%F A122868 a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k,k)*3^(n-k).
%F A122868 E.g.f. : exp(3x)*Bessel_I(0,2*sqrt(3)x).
%F A122868 D-finite with recurrence: n*a(n) + 3*(1-2*n)*a(n-1) + 3*(1-n)*a(n-2) = 0. - _R. J. Mathar_, Nov 14 2011 [proved in Belbachir et al. (see Table 1)]
%F A122868 a(n) ~ (1+sqrt(3))*(3+2*sqrt(3))^n/(2*sqrt(2*Pi*n)). - _Vaclav Kotesovec_, Oct 19 2012
%F A122868 a(n) = (-3)^(n/2)*LegendreP(n, -(-3)^(1/2)). - _Mark van Hoeij_, Nov 13 2022
%p A122868 b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
%p A122868       `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
%p A122868       `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
%p A122868     end:
%p A122868 a:= n-> b(n, 0):
%p A122868 seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 21 2021
%p A122868 r := (-3)^(1/2): seq(simplify(r^n*LegendreP(n, -r)), n=0..10); # _Mark van Hoeij_, Nov 13 2022
%t A122868 CoefficientList[Series[1/Sqrt[1-6*x-3*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%o A122868 (Maxima) a(n):=coeff(expand((1+3*x+3*x^2)^n),x,n);
%o A122868 makelist(a(n),n,0,12);
%o A122868 (PARI) my(x = 'x + O('x^30)); Vec(1/sqrt(1-6*x-3*x^2)) \\ _Michel Marcus_, Jan 29 2016
%Y A122868 Top row of array in A232973.
%K A122868 nonn,easy
%O A122868 0,2
%A A122868 _Paul Barry_, Sep 16 2006