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A370890 A(n, k) = 2^n*Pochhammer(k/2, floor((n+1)/2)). Square array read by ascending antidiagonals.

%I A370890 #11 Mar 06 2024 07:59:35
%S A370890 1,0,1,0,1,1,0,2,2,1,0,6,4,3,1,0,12,16,6,4,1,0,60,32,30,8,5,1,0,120,
%T A370890 192,60,48,10,6,1,0,840,384,420,96,70,12,7,1,0,1680,3072,840,768,140,
%U A370890 96,14,8,1,0,15120,6144,7560,1536,1260,192,126,16,9,1
%N A370890 A(n, k) = 2^n*Pochhammer(k/2, floor((n+1)/2)). Square array read by ascending antidiagonals.
%H A370890 Paolo Xausa, <a href="/A370890/b370890.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals, flattened).
%e A370890 The array starts:
%e A370890 [0] 1,  1,   1,   1,   1,    1,    1,    1,    1,    1, ...
%e A370890 [1] 0,  1,   2,   3,   4,    5,    6,    7,    8,    9, ...
%e A370890 [2] 0,  2,   4,   6,   8,   10,   12,   14,   16,   18, ...
%e A370890 [3] 0,  6,  16,  30,  48,   70,   96,  126,  160,  198, ...
%e A370890 [4] 0, 12,  32,  60,  96,  140,  192,  252,  320,  396, ...
%e A370890 [5] 0, 60, 192, 420, 768, 1260, 1920, 2772, 3840, 5148, ...
%e A370890 .
%e A370890 Seen as the triangle T(n, k) = A(n - k, k):
%e A370890 [0] 1;
%e A370890 [1] 0,   1;
%e A370890 [2] 0,   1,   1;
%e A370890 [3] 0,   2,   2,  1;
%e A370890 [4] 0,   6,   4,  3,  1;
%e A370890 [5] 0,  12,  16,  6,  4,  1;
%e A370890 [6] 0,  60,  32, 30,  8,  5, 1;
%e A370890 [7] 0, 120, 192, 60, 48, 10, 6, 1;
%p A370890 A := (n, k) -> 2^n*pochhammer(k/2, iquo(n+1,2)):
%p A370890 for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
%p A370890 T := (n, k) -> A(n - k, k):
%p A370890 seq(seq(T(n, k), k = 0..n), n = 0..10);
%t A370890 A370890[n_, k_] := 2^n*Pochhammer[k/2, Floor[(n+1)/2]];
%t A370890 Table[A370890[n-k, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Mar 06 2024 *)
%o A370890 (SageMath)  # Note the use of different kinds of division.
%o A370890 def A(n, k): return 2**n * rising_factorial(k/2, (n+1)//2)
%o A370890 for n in range(0, 9): print([A(n, k) for k in range(0, 9)])
%Y A370890 Rows: A000012, A001477, A005843, A054000, A134582.
%Y A370890 Columns: A000007, A081125, A355989.
%Y A370890 Cf. A370419.
%K A370890 nonn,tabl
%O A370890 0,8
%A A370890 _Peter Luschny_, Mar 04 2024