OFFSET
1,2
COMMENTS
Principal diagonal: A007585
Antidiagonal sums: A002417
row 1, (1,2,3,4,5,...)**(1,3,5,7,9,...): A000330
row 2, (1,2,3,4,5,...)**(3,5,7,9,...): A051925
row 3, (1,2,3,4,5,...)**(5,7,9,11,...): (2*k^3 + 15*k^2 + 13*k)/6
row 4, (1,2,3,4,5,...)**(7,9,11,13,...): (2*k^3 + 21*k^2 + 19*k)/6
For a guide to related arrays, see A213500.
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) = (2*n - 1) - (2*n - 3)*x and g(x) = (1 - x )^4.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1....5....14...30....55....91
3....11...26...50....85....133
5....17...38...70....115...175
7....23...50...90....145...217
9....29...62...110...175...259
11...35...74...130...205...301
MATHEMATICA
b[n_] := n; c[n_] := 2 n - 1;
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}] (* A213750 *)
d = Table[t[n, n], {n, 1, 40}] (* A007585 *)
s1 = Table[s[n], {n, 1, 50}] (* A002417 *)
FindLinearRecurrence[s1]
FindGeneratingFunction[s1, x]
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jun 20 2012
STATUS
approved