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A362378 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 * (j+1)^(n-2*j-1) / (j! * (n-3*j)!).

%I A362378 #20 Feb 16 2025 08:34:05
%S A362378 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,9,1,1,1,1,4,17,41,1,1,1,1,5,25,
%T A362378 81,191,1,1,1,1,6,33,121,441,1191,1,1,1,1,7,41,161,751,3641,9353,1,1,
%U A362378 1,1,8,49,201,1121,7351,33825,77897,1
%N A362378 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 * (j+1)^(n-2*j-1) / (j! * (n-3*j)!).
%H A362378 Seiichi Manyama, <a href="/A362378/b362378.txt">Antidiagonals n = 0..139, flattened</a>
%H A362378 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362378 E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^3/6 * A_k(x)).
%F A362378 A_k(x) = exp(x - LambertW(-k*x^3/6 * exp(x))).
%F A362378 A_k(x) = -6 * LambertW(-k*x^3/6 * exp(x))/(k*x^3) for k > 0.
%e A362378 Square array begins:
%e A362378   1,   1,   1,   1,    1,    1,    1, ...
%e A362378   1,   1,   1,   1,    1,    1,    1, ...
%e A362378   1,   1,   1,   1,    1,    1,    1, ...
%e A362378   1,   2,   3,   4,    5,    6,    7, ...
%e A362378   1,   9,  17,  25,   33,   41,   49, ...
%e A362378   1,  41,  81, 121,  161,  201,  241, ...
%e A362378   1, 191, 441, 751, 1121, 1551, 2041, ...
%o A362378 (PARI) T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(j+1)^(n-2*j-1)/(j!*(n-3*j)!));
%Y A362378 Columns k=0..3 give A000012, A362381, A362390, A362391.
%Y A362378 Cf. A362043, A362377.
%K A362378 nonn,tabl
%O A362378 0,14
%A A362378 _Seiichi Manyama_, Apr 20 2023