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%I #16 Apr 16 2026 17:13:53
%S 1,1,0,1,3,0,1,10,45,0,1,21,256,1743,0,1,36,849,13840,125625,0,1,55,
%T 2136,61941,1282816,14554683,0,1,78,4525,204576,7214529,181649920,
%U 2473184805,0,1,105,8520,554755,29419776,1229315061,36478744576,579439207623,0
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where A(n,k) = (2*n)! * [x^(2*n)] (f(x)^k + f(-x)^k)/2 and f(x) = 1/(cosh(x) - sqrt(2) * sinh(x)).
%H Paolo Xausa, <a href="/A395200/b395200.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals, flattened).
%F A(0,k) = 1 and A(n,k) = k*(k+1) * A(n-1,k+2) + k^2 * A(n-1,k) for n > 0.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 3, 10, 21, 36, 55, ...
%e 0, 45, 256, 849, 2136, 4525, ...
%e 0, 1743, 13840, 61941, 204576, 554755, ...
%e 0, 125625, 1282816, 7214529, 29419776, 97032025, ...
%e 0, 14554683, 181649920, 1229315061, 5958690816, 23118264655, ...
%t A395200[n_, k_] := A395200[n, k] = If[n == 0, 1, k*(k+1)*A395200[n-1, k+2] + k^2*A395200[n-1, k]];
%t Table[A395200[k, n-k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Apr 16 2026 *)
%o (PARI) a(n, k) = if(n==0, 1, k*(k+1)*a(n-1, k+2)+k^2*a(n-1, k));
%Y Columns k=0..3 give A000007, |A012494(n)|, |A024293(n+1)|, A395202.
%Y Cf. A341268, A395201.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, Apr 15 2026