OFFSET
1,2
COMMENTS
Principal diagonal: A213772
Antidiagonal sums: A132117
Row 1, (1,4,7,10,...)**(1,2,3,4,...): A002411
Row 2, (1,4,7,10,...)**(2,3,4,5,...): A162260
Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 - 2*k)/2
Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 - 3*k)/2
For a guide to related arrays, see A212500.
LINKS
Clark Kimberling, Antidiagonals n = 1..60, flattened
FORMULA
T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(n + (n+1)*x - (n+2)*x^2) and g(x) = (1 - x)^4.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1....6....18...40....75....126
2....11...30...62....110...177
3....16...42...84....145...228
4....21...54...106...180...279
5....26...66...128...215...330
MATHEMATICA
b[n_]:=3n-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}] (* A213771 *)
Table[t[n, n], {n, 1, 40}] (* A213772 *)
s[n_]:=Sum[t[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A132117 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jul 04 2012
STATUS
approved