OFFSET
1,2
COMMENTS
Principal diagonal: A213562
Antidiagonal sums: A213563
Row 1, (1,4,9,...)**(1,3,6,...): A005585
Row 2, (1,4,9,...)**(3,6,10,...): (2*k^5 +25*k^4 + 120*k^3 + 155*k^2 + 58*k)/120
Row 3, (1,4,9,...)**(6,10,15,...): (2*k^5 +35*k^4 + 60*k^3 + 325*k^2 + 118*k)/120
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*(n + 1) - (n^2 - n - 2)*x - (n^2 + n - 2)*x^2 + n*(n - 1)*x^3 and g(x) = 2*(1 - x)^6.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1....7.....27....77....182
3....18....61....157...342
6....34....109...267...557
10...55....171...407...827
15...81....247...577...1152
21...112...337...777...1532
MATHEMATICA
b[n_] := n^2; c[n_] := n (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}] (* A213561 *)
d = Table[t[n, n], {n, 1, 40}] (* A213562 *)
s1 = Table[s[n], {n, 1, 50}] (* A213563 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jun 18 2012
STATUS
approved