OFFSET
1,2
COMMENTS
LINKS
Clark Kimberling, Antidiagonals n = 1..60, flattened
FORMULA
T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).
G.f. for row n: f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2n - 1)*x + ((n - 1)^2)*x^2 and g(x) = (1 - x)^6.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1....7.....27....77....182
4....21....67....167...357
9....43....127...297...602
16...73....207...467...917
25...111...307...677...1302
36...157...427...927...1757
MATHEMATICA
b[n_] := n (n + 1)/2; c[n_] := n^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}] (* A213564 *)
d = Table[t[n, n], {n, 1, 40}] (* A213565 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A101094 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jun 18 2012
STATUS
approved