OFFSET
1,2
COMMENTS
Principal diagonal: A213556.
Antidiagonal sums: A213547.
Row 1, (1,8,27,...)**(1,2,3,...): A024166.
Row 2, (1,8,27,...)**(2,3,4,...): (3*k^5 + 30*k^4 + 55*k^3 + 30*k^2 + 2*k)/60.
Row 3, (1,8,27,...)**(3,4,5,...): (3*k^5 + 45*k^4 + 85*k^3 + 45*k^2 + 2*k)/60.
For a guide to related arrays, see A213500.
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 + (3*n + 1)*x - (3*n - 4)*x^2 - (n - 1)*x^3 and g(x) = (1 - x)^6.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1...10...46....146...371....812
2...19...82....246...596....1253
3...28...118...346...821....1694
4...37...154...446...1046...2135
5...46...190...546...1271...2576
6...55...226...646...1496...3017
MATHEMATICA
b[n_] := n^3; 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}] (* A213555 *)
d = Table[t[n, n], {n, 1, 40}] (* A213556 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213547 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jun 17 2012
STATUS
approved