OFFSET
1,2
COMMENTS
LINKS
Clark Kimberling, Antidiagonals n = 1..60
FORMULA
T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).
G.f. for row n: f(x)/g(x), where f(x) = n*(n+1) - 2*((n-1)^2)*x + n*(n-1)*x^2 and g(x) = 2*(1 - x)^5.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
. 1, 5, 15, 35, 70, ...
. 3, 12, 31, 65, 120, ...
. 6, 22, 53, 105, 185, ...
. 10, 35, 81, 155, 265, ...
. 15, 51, 115, 215, 360, ...
. 21, 70, 155, 285, 470, ...
...
T(5,1) = (1)**(15) = 15;
T(5,2) = (1,2)**(15,21) = 1*21 + 2*15 = 51;
T(5,3) = (1,2,3)**(15,21,28) = 1*28 + 2*21 + 3*15 = 115;
T(4,4) = (1,2,3,4)**(10,15,21,28) = 1*28 + 2*21 + 3*15 + 4*10 = 155.
MATHEMATICA
b[n_] := n; 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}] (* A213548 *)
d = Table[t[n, n], {n, 1, 40}] (* A213549 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A051836 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jun 16 2012
STATUS
approved