A232973 -id:A232973 - cp's OEIS frontend

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

1, 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201, 1254816463923, 7873205412825, 49482344889261, 311457546052659, 1963051327342449, 12387750763156227, 78258731003169435

Offset: 0

Paul Barry, Sep 16 2006

Binomial transform of A084609. Central coefficients of (1+3x+3x^2)^n.
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
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

Alois P. Heinz, Table of n, a(n) for n = 0..1236 (first 201 terms from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
W. Y. C. Chen, N. Y. Li, L. W. Shapiro and S. H. F. Yan, Matrix identities on weighted partial Motzkin paths, European J. Combinatorics, 28 (2007), 1196-2007.
M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, Volume 30, Issue 6, pp 1427-1452.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
J. L. Ramírez and V. F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.

Top row of array in A232973.

b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
      `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
      `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 21 2021
r := (-3)^(1/2): seq(simplify(r^n*LegendreP(n, -r)), n=0..10); # Mark van Hoeij, Nov 13 2022

CoefficientList[Series[1/Sqrt[1-6*x-3*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

a(n):=coeff(expand((1+3*x+3*x^2)^n),x,n);
makelist(a(n),n,0,12);

my(x = 'x + O('x^30)); Vec(1/sqrt(1-6*x-3*x^2)) \\ Michel Marcus, Jan 29 2016

a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k,k)*3^(n-k).
E.g.f. : exp(3x)*Bessel_I(0,2*sqrt(3)x).
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)]
a(n) ~ (1+sqrt(3))*(3+2*sqrt(3))^n/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 19 2012
a(n) = (-3)^(n/2)*LegendreP(n, -(-3)^(1/2)). - Mark van Hoeij, Nov 13 2022

A232969 The sequence S(n,n) that enumerates a certain class of lattice paths from (0,0) to (n,n).

Original entry on oeis.org

1, 6, 60, 675, 7992, 97416, 1209951, 15227190, 193507056, 2477564820, 31910429520, 412987306320, 5366341375695, 69965422235442, 914825583252396, 11991475839917115, 157524763370404320, 2073261181622482080, 27333449595845251524, 360903785815145617992

Offset: 0

N. J. A. Sloane, Dec 05 2013

nonn

See Dziemianczuk for precise definition.

Alois P. Heinz, Table of n, a(n) for n = 0..884 (first 101 terms from Lars Blomberg)
M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.

Leading column of array in A232973.

b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
      `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
      `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Sep 21 2021

Table[Function[k, Sum[Sum[Binomial[k, j] Binomial[j, i - j] Binomial[2 k + n - i, k], {j, 0, i}], {i, 0, n + k}]]@ n, {n, 0, 19}] (* Michael De Vlieger, Jul 22 2017 *)

\\ Dziemianczuk, Proposition 1
S(n,k)=sum(i=0,n+k,sum(j=0,i,binomial(k,j)*binomial(j,i-j)*binomial(2*k+n-i,k)));
vector(20,x,x--;S(x,x)) \\ Lars Blomberg, Jul 20 2017

a(8)-a(19) from Lars Blomberg, Jul 20 2017

cp's OEIS Frontend

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

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