OFFSET
1,2
COMMENTS
LINKS
Clark Kimberling, Antidiagonals n = 1..60, flattened
Henri Muehle, Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details, arXiv preprint arXiv:1301.1654, 2013.
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^2 - (n^2 - 2*n - 1)*x - (n^2 - 2)*x^2 - ((n - 1)^2)*x^3 and g(x) = (1 - x)^6.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1....8.....34....104...259....560
4....25....88....234...524....1043
9....52....170...424...899....1708
16...89....280...674...1384...2555
25...136...418...984...1979...3584
...
T(5,1) = (1)**(25) = 25
T(5,2) = (1,4)**(25,36) = 1*36+4*25 = 136
T(5,3) = (1,4,9)**(25,36,49) = 1*49+4*36+9*25 = 418
MATHEMATICA
b[n_] := n^2; c[n_] := n^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}] (* A213505 *)
d = Table[t[n, n], {n, 1, 40}] (* A213546 *)
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 16 2012
STATUS
approved