A257365 Triangle, read by rows, T(n,k) = Sum_{m=0..(n-k)/2} C(k,m)*C(n-2*m,k).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 8, 16, 13, 5, 1, 1, 10, 28, 32, 19, 6, 1, 1, 12, 44, 68, 55, 26, 7, 1, 1, 14, 64, 128, 136, 86, 34, 8, 1, 1, 16, 88, 220, 296, 241, 126, 43, 9, 1, 1, 18, 116, 352, 584, 592, 393, 176, 53, 10, 1

Offset: 0

Author

Vladimir Kruchinin, Apr 21 2015

Keywords

• nonn
• tabl

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)

Examples

			1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 6, 8, 4, 1;
1, 8, 16, 13, 5, 1;
		

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.

Crossrefs

Cf. A006139.

Programs

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 *)

Maxima

T(n,k):=sum(binomial(k,m)*binomial(n-2*m,k),m,0,(n-k)/2);

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)