%I #3 Jun 07 2021 00:42:13
%S 1,2,4,8,2,16,16,32,80,64,320,40,128,1120,560,256,3584,4480,512,10752,
%T 26880,2240,1024,30720,134400,44800,2048,84480,591360,492800,4096,
%U 225280,2365440,3942400,246400,8192,585728,8785920,25625600,6406400
%N T(n, k) = (3^(-k)*n!*2^(n - 3*k))/(k!*(n - 3*k)!), for n >= 0 and 0 <= k <= floor(n/3). Triangle read by rows.
%H mjqxxxx, <a href="https://math.stackexchange.com/q/4164050">Proof of conjectured formulas for A336614</a>, Mathematics Stack Exchange.
%e [ 0] 1;
%e [ 1] 2;
%e [ 2] 4;
%e [ 3] 8, 2;
%e [ 4] 16, 16;
%e [ 5] 32, 80;
%e [ 6] 64, 320, 40;
%e [ 7] 128, 1120, 560;
%e [ 8] 256, 3584, 4480;
%e [ 9] 512, 10752, 26880, 2240;
%e [10] 1024, 30720, 134400, 44800;
%e [11] 2048, 84480, 591360, 492800;
%e [12] 4096, 225280, 2365440, 3942400, 246400.
%p t := (n, k) -> k^n*n!: s := (n, k) -> 2^(3*k)*(n - 3*k)!:
%p T := (n, k) -> t(n, 2) / (t(k, 3) * s(n, k)):
%p seq(lprint([n], seq(T(n, k), k = 0..n/3)), n = 0..12);
%Y A336614 (row sums).
%Y Cf. A344914.
%K nonn,tabf
%O 0,2
%A _Peter Luschny_, Jun 06 2021