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

0,5

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

Columns k=0..5 give A331551, A331552, A000290(n+1), A002457, A108666(n+1), A331323.

T(n,n+3) gives A331512.

Cf. A307883, A331513, A331514.

Sequence in context: A228342 A027739 A193307 * A201050 A299321 A357097

Adjacent sequences: A331508 A331509 A331510 * A331512 A331513 A331514

Keyword

sign,tabl

Author

Seiichi Manyama, Jan 18 2020

Status

approved