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%I #23 Apr 16 2023 09:48:49
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,-1,-3,1,1,1,1,-2,-7,-9,1,1,1,1,
%T -3,-11,-19,-9,1,1,1,1,-4,-15,-29,1,36,1,1,1,1,-5,-19,-39,31,211,225,
%U 1,1,1,1,-6,-23,-49,81,526,1009,477,1,1,1,1,-7,-27,-59,151,981,2353,953,-819,1
%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (-k/6)^j * binomial(n-2*j,j)/(n-2*j)!.
%H Seiichi Manyama, <a href="/A362302/b362302.txt">Antidiagonals n = 0..139, flattened</a>
%F E.g.f. of column k: exp(x - k*x^3/6).
%F T(n,k) = T(n-1,k) - k * binomial(n-1,2) * T(n-3,k) for n > 2.
%F T(n,k) = n! * Sum_{j=0..floor(n/3)} (-k/6)^j / (j! * (n-3*j)!).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 0, -1, -2, -3, -4, -5, ...
%e 1, -3, -7, -11, -15, -19, -23, ...
%e 1, -9, -19, -29, -39, -49, -59, ...
%e 1, -9, 1, 31, 81, 151, 241, ...
%o (PARI) T(n, k) = n!*sum(j=0, n\3, (-k/6)^j/(j!*(n-3*j)!));
%Y Columns k=0..2 give A000012, A351929, A362309.
%Y Main diagonal gives A362303.
%Y T(n,2*n) gives A362304.
%Y T(n,6*n) gives A362305.
%Y Cf. A362043, A362277.
%K sign,tabl
%O 0,20
%A _Seiichi Manyama_, Apr 15 2023