OFFSET
0,8
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6
FORMULA
a(n,k) = Sum_{j=0..k} C(2k,2j) c(j) a(n-1,k-j) where c(j) = C(2j,j)/(j+1) = A000108(j) with a(0,0) = 1 and a(0,k) = 0 for k>0.
EXAMPLE
Rows start:
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 5, 14, 42, 132, ...
1, 2, 10, 70, 588, 5544, 56628, ...
1, 3, 24, 285, 4242, 73206, 1403028, ...
MAPLE
a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
add(binomial(2*k, 2*j)*binomial(2*j, j)/
(j+1)*a(n-1, k-j), j=0..k))
end:
seq(seq(a(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, May 06 2014
MATHEMATICA
a[n_, k_] := a[n, k] = If[n == 0, If[k == 0, 1, 0], Sum[Binomial[2*k, 2*j]* Binomial[2*j, j]/(j+1)*a[n-1, k-j], {j, 0, k}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Aug 23 2001
STATUS
approved