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A(n, k) = 2^n*Pochhammer(k/2, floor((n+1)/2)). Square array read by ascending antidiagonals.
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%I #11 Mar 06 2024 07:59:35

%S 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 192,60,48,10,6,1,0,840,384,420,96,70,12,7,1,0,1680,3072,840,768,140,

%U 96,14,8,1,0,15120,6144,7560,1536,1260,192,126,16,9,1

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

%H Paolo Xausa, <a href="/A370890/b370890.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals, flattened).

%e The array starts:

%e [0] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e [1] 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

%e [2] 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, ...

%e [3] 0, 6, 16, 30, 48, 70, 96, 126, 160, 198, ...

%e [4] 0, 12, 32, 60, 96, 140, 192, 252, 320, 396, ...

%e [5] 0, 60, 192, 420, 768, 1260, 1920, 2772, 3840, 5148, ...

%e .

%e Seen as the triangle T(n, k) = A(n - k, k):

%e [0] 1;

%e [1] 0, 1;

%e [2] 0, 1, 1;

%e [3] 0, 2, 2, 1;

%e [4] 0, 6, 4, 3, 1;

%e [5] 0, 12, 16, 6, 4, 1;

%e [6] 0, 60, 32, 30, 8, 5, 1;

%e [7] 0, 120, 192, 60, 48, 10, 6, 1;

%p A := (n, k) -> 2^n*pochhammer(k/2, iquo(n+1,2)):

%p for n from 0 to 5 do seq(A(n, k), k = 0..9) od;

%p T := (n, k) -> A(n - k, k):

%p seq(seq(T(n, k), k = 0..n), n = 0..10);

%t A370890[n_, k_] := 2^n*Pochhammer[k/2, Floor[(n+1)/2]];

%t Table[A370890[n-k, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Mar 06 2024 *)

%o (SageMath) # Note the use of different kinds of division.

%o def A(n, k): return 2**n * rising_factorial(k/2, (n+1)//2)

%o for n in range(0, 9): print([A(n, k) for k in range(0, 9)])

%Y Rows: A000012, A001477, A005843, A054000, A134582.

%Y Columns: A000007, A081125, A355989.

%Y Cf. A370419.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Mar 04 2024