|
|
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, 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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
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).
Central coefficients = A006139. (End)
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 1/(1-y-x*(1+y^2)).
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
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|