OFFSET
1,2
COMMENTS
LINKS
Clark Kimberling, Antidiagonals n = 1..60, flattened
FORMULA
T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(ceiling(n/2) + m(n)*x - floor(n/2)*x^2), where m(n) = (n+1 mod 2), and g(x) = (1+x)^2 *(1-x)^4.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1...2...5....8....14...20...30...40
1...3...6....11...17...26...36...50
2...4...9....14...23...32...46...60
2...5...10...17...26...38...52...70
3...6...13...20...32...44...62...80
MATHEMATICA
b[n_]:=Floor[(n+1)/2]; c[n_]:=Floor[(n+1)/2];
t[n_, k_]:=Sum[b[k-i]c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:=Table[t[n, k], {k, 1, 60}] (* A213849 *)
d=Table[t[n, n], {n, 1, 50}] (* A049778 *)
s[n_]:=Sum[t[i, n+1-i], {i, 1, n}]
s1=Table[s[n], {n, 1, 50}] (* A213850 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jul 05 2012
STATUS
approved