A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)*x)/(1 - 2*(k-1)*x + ((k-3)*x)^2)^(3/2).
1, 1, 0, 1, 2, -15, 1, 4, -6, 32, 1, 6, 9, -12, 105, 1, 8, 30, 16, 30, -576, 1, 10, 57, 140, 25, 60, 105, 1, 12, 90, 384, 630, 36, -140, 5760, 1, 14, 129, 772, 2505, 2772, 49, -280, -13167, 1, 16, 174, 1328, 6430, 16008, 12012, 64, 630, -30400
Offset: 0
Examples
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)
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Crossrefs
Programs
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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
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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 *)
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