OFFSET
0,5
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
T(n,k) = Sum_{j=0..n} (k-3)^(n-j) * (n+j+1) * binomial(n,j) * binomial(n+j,j).
T(n,k) = Sum_{j=0..n} (k-2)^j * (j+1) * binomial(n+1,j+1)^2.
T(n,k) = (n + 1)^2*hypergeom([-n, -n], [2], k - 2). - Peter Luschny, Jan 20 2020
n * (2*n-1) * T(n,k) = 2 * (2 * (k-1) * n^2 - k + 2) * T(n-1,k) - (k-3)^2 * n * (2*n+1) * T(n-2,k) for n>1. - Seiichi Manyama, Jan 25 2020
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, ...
-15, -6, 9, 30, 57, 90, ...
32, -12, 16, 140, 384, 772, ...
105, 30, 25, 630, 2505, 6430, ...
-576, 60, 36, 2772, 16008, 52524, ...
.
From Peter Luschny, Jan 20 2020: (Start)
Read by ascending antidiagonals gives:
[0] 1
[1] 0, 1
[2] -15, 2, 1
[3] 32, -6, 4, 1
[4] 105, -12, 9, 6, 1
[5] -576, 30, 16, 30, 8, 1
[6] 105, 60, 25, 140, 57, 10, 1
[7] 5760, -140, 36, 630, 384, 90, 12, 1
[8] -13167, -280, 49, 2772, 2505, 772, 129, 14, 1
[9] -30400, 630, 64, 12012, 16008, 6430, 1328, 174, 16, 1 (End)
MAPLE
T := (n, k) -> (n + 1)^2*hypergeom([-n, -n], [2], k - 2):
seq(lprint(seq(simplify(T(n, k)), k=0..7)), n=0..6) # Peter Luschny, Jan 20 2020
MATHEMATICA
T[n_, k_] := (n + 1)^2 * HypergeometricPFQ[{-n, -n}, {2}, k - 2]; Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Jan 18 2020
STATUS
approved