OFFSET
1,2
COMMENTS
LINKS
Clark Kimberling, Antidiagonals n = 1..80, flattened
FORMULA
T(n,k) = 2*T(n,k-1) -T(n,k-2) - 4*T(n,k-3) +T(n,k-4) +2*T(n,k-5) -T(n,k-6).
G.f. for row n: f(x)/g(x), where f(x) = [(n+1)/2] + [(n+2)/2]*x + ([(n-1)/2] + [(n+1)/2])*x^2 - (1+[n/2]-(n mod 2))*x^3 + [n/2]*x^4 and g(x) = (1 + x)^2 *(1 - x)^4, where [ ] = floor.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1...3....6....11...17...26...36....50
1...4....8....14...22...32...45....60
2...6....11...19...28...41...55....74
2...7....13...22...33...47...64....84
3...9....16...27...39...56...74....98
3...10...18...30...44...62...83....108
4...12...21...35...50...71...93....122
4...13...23...38...55...77...102...132
MATHEMATICA
b[n_] := Floor[(n + 2)/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}] (* A213783 *)
Table[t[n, n], {n, 1, 40}] (* A213759 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213760 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jun 22 2012
STATUS
approved