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A362483 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 * (2*j+1)^(n-j-1) / (j! * (n-2*j)!).

%I A362483 #14 Feb 16 2025 08:34:05
%S A362483 1,1,1,1,1,1,1,1,2,1,1,1,3,10,1,1,1,4,19,70,1,1,1,5,28,169,646,1,1,1,
%T A362483 6,37,298,2041,7576,1,1,1,7,46,457,4186,30811,106744,1,1,1,8,55,646,
%U A362483 7081,74116,560827,1761628,1,1,1,9,64,865,10726,141901,1578340,11957905,33361948,1
%N A362483 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 * (2*j+1)^(n-j-1) / (j! * (n-2*j)!).
%H A362483 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362483 E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^2/2 * A_k(x)^2).
%F A362483 A_k(x) = exp(x - LambertW(-k*x^2 * exp(2*x))/2).
%F A362483 A_k(x) = sqrt( -LambertW(-k*x^2 * exp(2*x))/(k*x^2) ) for k > 0.
%e A362483 Square array begins:
%e A362483   1,   1,    1,    1,    1,     1, ...
%e A362483   1,   1,    1,    1,    1,     1, ...
%e A362483   1,   2,    3,    4,    5,     6, ...
%e A362483   1,  10,   19,   28,   37,    46, ...
%e A362483   1,  70,  169,  298,  457,   646, ...
%e A362483   1, 646, 2041, 4186, 7081, 10726, ...
%o A362483 (PARI) T(n, k) = n! * sum(j=0, n\2, (k/2)^j*(2*j+1)^(n-j-1)/(j!*(n-2*j)!));
%Y A362483 Columns k=0..3 give A000012, A362474, A143768, A362475.
%Y A362483 Cf. A362377, A362490.
%K A362483 nonn,tabl
%O A362483 0,9
%A A362483 _Seiichi Manyama_, Apr 21 2023