A307910 - OEIS

%I #21 May 07 2019 11:00:21

%S 1,1,0,1,1,0,1,2,3,0,1,3,8,7,0,1,4,15,32,19,0,1,5,24,81,136,51,0,1,6,

%T 35,160,459,592,141,0,1,7,48,275,1120,2673,2624,393,0,1,8,63,432,2275,

%U 8064,15849,11776,1107,0,1,9,80,637,4104,19375,59136,95175,53344,3139,0

%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*k*x + k*(k-4)*x^2).

%H Seiichi Manyama, <a href="/A307910/b307910.txt">Antidiagonals n = 0..139, flattened</a>

%F A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + k*x^2)^n.

%F A(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j) * binomial(2*j,j).

%F n * A(n,k) = k * (2*n-1) * A(n-1,k) - k * (k-4) * (n-1) * A(n-2,k).

%e Square array begins:

%e    1,   1,     1,     1,      1,       1,       1, ...

%e    0,   1,     2,     3,      4,       5,       6, ...

%e    0,   3,     8,    15,     24,      35,      48, ...

%e    0,   7,    32,    81,    160,     275,     432, ...

%e    0,  19,   136,   459,   1120,    2275,    4104, ...

%e    0,  51,   592,  2673,   8064,   19375,   40176, ...

%e    0, 141,  2624, 15849,  59136,  168125,  400896, ...

%e    0, 393, 11776, 95175, 439296, 1478125, 4053888, ...

%t A[n_, k_] := k^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 4/k]; A[0, _] = 1; A[_, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, May 07 2019 *)

%Y Columns k=0..4 give A000007, A002426, A006139, A122868, A059304.

%Y Main diagonal gives A092366.

%Y Cf. A107267, A292627, A307819, A307847, A307855, A307883.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, May 05 2019