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A362490 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 * (3*j+1)^(n-2*j-1) / (j! * (n-3*j)!).

%I A362490 #20 Feb 16 2025 08:34:05
%S A362490 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,17,1,1,1,1,4,33,161,1,1,1,1,5,
%T A362490 49,321,1351,1,1,1,1,6,65,481,2841,12391,1,1,1,1,7,81,641,4471,31641,
%U A362490 153385,1,1,1,1,8,97,801,6241,57751,498849,2388905,1
%N A362490 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 * (3*j+1)^(n-2*j-1) / (j! * (n-3*j)!).
%H A362490 Winston de Greef, <a href="/A362490/b362490.txt">Table of n, a(n) for n = 0..11324</a> (150 antidiagonals)
%H A362490 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362490 E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^3/6 * A_k(x)^3).
%F A362490 A_k(x) = exp(x - LambertW(-k*x^3/2 * exp(3*x))/3).
%F A362490 A_k(x) = ( -2 * LambertW(-k*x^3/2 * exp(3*x))/(k*x^3) )^(1/3) for k > 0.
%e A362490 Square array begins:
%e A362490   1,    1,    1,    1,    1,    1,     1, ...
%e A362490   1,    1,    1,    1,    1,    1,     1, ...
%e A362490   1,    1,    1,    1,    1,    1,     1, ...
%e A362490   1,    2,    3,    4,    5,    6,     7, ...
%e A362490   1,   17,   33,   49,   65,   81,    97, ...
%e A362490   1,  161,  321,  481,  641,  801,   961, ...
%e A362490   1, 1351, 2841, 4471, 6241, 8151, 10201, ...
%o A362490 (PARI) T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(3*j+1)^(n-2*j-1)/(j!*(n-3*j)!));
%Y A362490 Columns k=0..3 give A000012, A362477, A362478, A362479.
%Y A362490 Cf. A362378.
%K A362490 nonn,tabl
%O A362490 0,14
%A A362490 _Seiichi Manyama_, Apr 22 2023