OFFSET
0,5
COMMENTS
From Emanuele Munarini, Feb 21 2017: (Start)
T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps X=(1,0), D=(1,1) and E=(3,1).
Row sums = A008998.
Central coefficients = A006139. (End)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
James East, Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
FORMULA
G.f.: 1/(1-y-x*(1+y^2)).
From Emanuele Munarini, Feb 21 2017: (Start)
G.f. for the triangle: 1/(1-x-x*y-x^3*y).
Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n,k). (End)
EXAMPLE
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 6, 8, 4, 1;
1, 8, 16, 13, 5, 1;
MATHEMATICA
Table[Sum[Binomial[k, m] Binomial[n - 2 m, k], {m, 0, (n - k)/2}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 21 2015 *)
PROG
(Maxima) T(n, k):=sum(binomial(k, m)*binomial(n-2*m, k), m, 0, (n-k)/2);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Apr 21 2015
STATUS
approved