cp's OEIS Frontend

A362394 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 A362394 #20 Feb 16 2025 08:34:05
%S A362394 1,1,1,1,1,1,1,1,0,1,1,1,-1,-5,1,1,1,-2,-11,-14,1,1,1,-3,-17,-11,56,1,
%T A362394 1,1,-4,-23,10,381,736,1,1,1,-5,-29,49,976,2461,1114,1,1,1,-6,-35,106,
%U A362394 1841,3736,-21083,-45156,1,1,1,-7,-41,181,2976,3121,-106910,-449623,-428660,1
%N A362394 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 A362394 Seiichi Manyama, <a href="/A362394/b362394.txt">Antidiagonals n = 0..139, flattened</a>
%H A362394 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362394 E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x - k*x^2/2 * A_k(x)).
%F A362394 A_k(x) = exp(x - LambertW(k*x^2/2 * exp(x))).
%F A362394 A_k(x) = 2 * LambertW(k*x^2/2 * exp(x))/(k*x^2) for k > 0.
%e A362394 Square array begins:
%e A362394   1,   1,    1,    1,    1,    1,     1, ...
%e A362394   1,   1,    1,    1,    1,    1,     1, ...
%e A362394   1,   0,   -1,   -2,   -3,   -4,    -5, ...
%e A362394   1,  -5,  -11,  -17,  -23,  -29,   -35, ...
%e A362394   1, -14,  -11,   10,   49,  106,   181, ...
%e A362394   1,  56,  381,  976, 1841, 2976,  4381, ...
%e A362394   1, 736, 2461, 3736, 3121, -824, -9539, ...
%o A362394 (PARI) T(n, k) = n! * sum(j=0, n\2, (-k/2)^j*(j+1)^(n-j-1)/(j!*(n-2*j)!));
%Y A362394 Columns k=0..3 give A000012, A362395, A362396, A362397.
%Y A362394 Cf. A362277, A362377.
%K A362394 sign,tabl
%O A362394 0,14
%A A362394 _Seiichi Manyama_, Apr 20 2023