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A362377 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (j+1)^(n-j-1) / (j! * (n-2*j)!).

%I A362377 #22 Feb 16 2025 08:34:05
%S A362377 1,1,1,1,1,1,1,1,2,1,1,1,3,7,1,1,1,4,13,34,1,1,1,5,19,85,216,1,1,1,6,
%T A362377 25,154,701,1696,1,1,1,7,31,241,1456,7261,15898,1,1,1,8,37,346,2481,
%U A362377 18136,89125,173468,1,1,1,9,43,469,3776,35761,260002,1277865,2161036,1
%N A362377 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (j+1)^(n-j-1) / (j! * (n-2*j)!).
%H A362377 Seiichi Manyama, <a href="/A362377/b362377.txt">Antidiagonals n = 0..139, flattened</a>
%H A362377 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362377 E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^2/2 * A_k(x)).
%F A362377 A_k(x) = exp(x - LambertW(-k*x^2/2 * exp(x))).
%F A362377 A_k(x) = -2 * LambertW(-k*x^2/2 * exp(x))/(k*x^2) for k > 0.
%e A362377 Square array begins:
%e A362377   1,    1,    1,     1,     1,     1,     1, ...
%e A362377   1,    1,    1,     1,     1,     1,     1, ...
%e A362377   1,    2,    3,     4,     5,     6,     7, ...
%e A362377   1,    7,   13,    19,    25,    31,    37, ...
%e A362377   1,   34,   85,   154,   241,   346,   469, ...
%e A362377   1,  216,  701,  1456,  2481,  3776,  5341, ...
%e A362377   1, 1696, 7261, 18136, 35761, 61576, 97021, ...
%o A362377 (PARI) T(n, k) = n! * sum(j=0, n\2, (k/2)^j*(j+1)^(n-j-1)/(j!*(n-2*j)!));
%Y A362377 Columns k=0..3 give A000012, A143740, A125500, A362380.
%Y A362377 Cf. A362378, A362394.
%K A362377 nonn,tabl
%O A362377 0,9
%A A362377 _Seiichi Manyama_, Apr 20 2023