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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).

%I A331511 #43 Jan 26 2020 01:13:04
%S A331511 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,
%T A331511 25,60,105,1,12,90,384,630,36,-140,5760,1,14,129,772,2505,2772,49,
%U A331511 -280,-13167,1,16,174,1328,6430,16008,12012,64,630,-30400
%N 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).
%H A331511 Seiichi Manyama, <a href="/A331511/b331511.txt">Antidiagonals n = 0..139, flattened</a>
%F A331511 T(n,k) = Sum_{j=0..n} (k-3)^(n-j) * (n+j+1) * binomial(n,j) * binomial(n+j,j).
%F A331511 T(n,k) = Sum_{j=0..n} (k-2)^j * (j+1) * binomial(n+1,j+1)^2.
%F A331511 T(n,k) = (n + 1)^2*hypergeom([-n, -n], [2], k - 2). - _Peter Luschny_, Jan 20 2020
%F A331511 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
%e A331511 Square array begins:
%e A331511       1,   1,  1,    1,     1,     1, ...
%e A331511       0,   2,  4,    6,     8,    10, ...
%e A331511     -15,  -6,  9,   30,    57,    90, ...
%e A331511      32, -12, 16,  140,   384,   772, ...
%e A331511     105,  30, 25,  630,  2505,  6430, ...
%e A331511    -576,  60, 36, 2772, 16008, 52524, ...
%e A331511 .
%e A331511 From _Peter Luschny_, Jan 20 2020: (Start)
%e A331511 Read by ascending antidiagonals gives:
%e A331511 [0]      1
%e A331511 [1]      0,    1
%e A331511 [2]    -15,    2,  1
%e A331511 [3]     32,   -6,  4,     1
%e A331511 [4]    105,  -12,  9,     6,     1
%e A331511 [5]   -576,   30, 16,    30,     8,    1
%e A331511 [6]    105,   60, 25,   140,    57,   10,    1
%e A331511 [7]   5760, -140, 36,   630,   384,   90,   12,   1
%e A331511 [8] -13167, -280, 49,  2772,  2505,  772,  129,  14,  1
%e A331511 [9] -30400,  630, 64, 12012, 16008, 6430, 1328, 174, 16, 1 (End)
%p A331511 T := (n, k) -> (n + 1)^2*hypergeom([-n, -n], [2], k - 2):
%p A331511 seq(lprint(seq(simplify(T(n,k)), k=0..7)), n=0..6) # _Peter Luschny_, Jan 20 2020
%t A331511 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 *)
%Y A331511 Columns k=0..5 give A331551, A331552, A000290(n+1), A002457, A108666(n+1), A331323.
%Y A331511 T(n,n+3) gives A331512.
%Y A331511 Cf. A307883, A331513, A331514.
%K A331511 sign,tabl
%O A331511 0,5
%A A331511 _Seiichi Manyama_, Jan 18 2020