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A338654 T(n, k) = 2^n * Product_{j=1..k} (j/2)^((-1)^(j - 1)). Triangle read by rows, for 0 <= k <= n.

%I A338654 #20 Dec 10 2023 17:33:24
%S A338654 1,2,1,4,2,2,8,4,4,6,16,8,8,12,6,32,16,16,24,12,30,64,32,32,48,24,60,
%T A338654 20,128,64,64,96,48,120,40,140,256,128,128,192,96,240,80,280,70,512,
%U A338654 256,256,384,192,480,160,560,140,630,1024,512,512,768,384,960,320,1120,280,1260,252
%N A338654 T(n, k) = 2^n * Product_{j=1..k} (j/2)^((-1)^(j - 1)). Triangle read by rows, for 0 <= k <= n.
%H A338654 Felix Fröhlich, <a href="/A338654/b338654.txt">Table of n, a(n) for n = 0..10000</a>
%e A338654 Triangle start:
%e A338654                              [0] 1
%e A338654                            [1] 2, 1
%e A338654                           [2] 4, 2, 2
%e A338654                         [3] 8, 4, 4, 6
%e A338654                       [4] 16, 8, 8, 12, 6
%e A338654                   [5] 32, 16, 16, 24, 12, 30
%e A338654                 [6] 64, 32, 32, 48, 24, 60, 20
%e A338654              [7] 128, 64, 64, 96, 48, 120, 40, 140
%e A338654          [8] 256, 128, 128, 192, 96, 240, 80, 280, 70
%e A338654      [9] 512, 256, 256, 384, 192, 480, 160, 560, 140, 630
%p A338654 T := (n, k) -> 2^n*mul((j/2)^((-1)^(j - 1)), j = 1 .. k):
%p A338654 seq(seq(T(n, k), k=0..n), n=0..9);
%p A338654 # Recurrence:
%p A338654 Trow := proc(n) if n = 0 then return [1] fi; Trow(n - 1);
%p A338654 n^irem(n, 2) * (4/n)^irem(n + 1, 2) * %[n]; [op(2 * %%), %] end:
%p A338654 seq(print(Trow(n)), n = 0..9);
%o A338654 (PARI) t(n, k) = 2^n * prod(j=1, k, ((j/2)^((-1)^(j - 1))))
%o A338654 trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
%o A338654 /* Print upper 10 rows of the triangle as follows: */
%o A338654 trianglerows(10) \\ _Felix Fröhlich_, Apr 22 2021
%Y A338654 T(n, 0) = A000079(n), T(n, n) = A056040(n), T(2*n, n) = A253665(n).
%Y A338654 Cf. A328002 (row sums), A163590.
%K A338654 nonn,tabl
%O A338654 0,2
%A A338654 _Peter Luschny_, Apr 22 2021