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A331514 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(1 - 2*k*x + ((k-2)*x)^2)^(3/2).

%I A331514 #43 Aug 21 2025 00:09:49
%S A331514 1,1,0,1,3,-6,1,6,6,0,1,9,30,10,30,1,12,66,140,15,0,1,15,114,450,630,
%T A331514 21,-140,1,18,174,1000,2955,2772,28,0,1,21,246,1850,8430,18963,12012,
%U A331514 36,630,1,24,330,3060,18855,69384,119812,51480,45,0
%N A331514 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(1 - 2*k*x + ((k-2)*x)^2)^(3/2).
%H A331514 Seiichi Manyama, <a href="/A331514/b331514.txt">Antidiagonals n = 0..139, flattened</a>
%F A331514 T(n,k) = (1/2) * Sum_{j=1..n+1} (k-2)^(n+1-j) * j * binomial(n+1,j) * binomial(n+1+j,j).
%F A331514 n * T(n,k) = k * (2*n+1) * T(n-1,k) - (k-2)^2 * (n+1) * T(n-2,k) for n > 1.
%F A331514 T(n,k) = ((n+2)/2) * Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
%F A331514 T(n,k) = Sum_{j=0..n} (k/2)^j * (-(k-2)^2/(2*k))^(n-j) * (2*j+1) * binomial(2*j,j) * binomial(j,n-j) for k > 0. - _Seiichi Manyama_, Aug 20 2025
%e A331514 Square array begins:
%e A331514     1,  1,    1,     1,     1,      1, ...
%e A331514     0,  3,    6,     9,    12,     15, ...
%e A331514    -6,  6,   30,    66,   114,    174, ...
%e A331514     0, 10,  140,   450,  1000,   1850, ...
%e A331514    30, 15,  630,  2955,  8430,  18855, ...
%e A331514     0, 21, 2772, 18963, 69384, 187425, ...
%t A331514 T[n_, k_] = 1/2 * Sum[If[k == 2 && n == j - 1, 1, (k - 2)^(n + 1 - j)] * j * Binomial[n + 1, j] * Binomial[n + 1 + j, j], {j, 1, n + 1}]; Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* _Amiram Eldar_, Jan 20 2020 *)
%o A331514 (PARI) T(n,k) = (1/2)*sum(j=1,n+1,(k-2)^(n+1-j)*j*binomial(n+1,j)*binomial(n+1+j,j));
%o A331514 matrix(7, 7, n, k, T(n-1, k-1)) \\ _Michel Marcus_, Jan 20 2020
%Y A331514 Columns k=1..5 give A000217(n+1), A002457, A002695(n+1), A331515, A331516.
%Y A331514 Cf. A307883, A331511.
%K A331514 sign,tabl
%O A331514 0,5
%A A331514 _Seiichi Manyama_, Jan 19 2020