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%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
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