OFFSET
1,2
COMMENTS
LINKS
Clark Kimberling, Antidiagonals n = 1..60, flattened
FORMULA
T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - 2*T(n,k-3) + 3*T(n,k-4) - T(n,k-5).
G.f. for row n: f(x)/g(x), where f(x) = x*(n + x - (2*n - 1)*x^2 + (n -1)*x^3) and g(x) = (1 + x)(1 - x)^4.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1...4....9....17...28...43....62
2...7....14...25...39...58....81
3...10...19...33...50...73....100
4...13...24...41...61...88....119
5...16...29...49...72...103...138
6...19...34...57...83...118...157
7...22...39...65...94...133...176
MATHEMATICA
b[n_] := Floor[(n + 2)/2]; c[n_] := n;
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}] (* A213781 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A005712 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jun 22 2012
STATUS
approved