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

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, 20, 128, 64, 64, 96, 48, 120, 40, 140, 256, 128, 128, 192, 96, 240, 80, 280, 70, 512, 256, 256, 384, 192, 480, 160, 560, 140, 630, 1024, 512, 512, 768, 384, 960, 320, 1120, 280, 1260, 252

Offset: 0

Author

Peter Luschny, Apr 22 2021

Keywords

• nonn
• tabl

Examples

			Triangle start:
                             [0] 1
                           [1] 2, 1
                          [2] 4, 2, 2
                        [3] 8, 4, 4, 6
                      [4] 16, 8, 8, 12, 6
                  [5] 32, 16, 16, 24, 12, 30
                [6] 64, 32, 32, 48, 24, 60, 20
             [7] 128, 64, 64, 96, 48, 120, 40, 140
         [8] 256, 128, 128, 192, 96, 240, 80, 280, 70
     [9] 512, 256, 256, 384, 192, 480, 160, 560, 140, 630
		

Links

• Felix Fröhlich, Table of n, a(n) for n = 0..10000

Crossrefs

T(n, 0) = A000079(n), T(n, n) = A056040(n), T(2*n, n) = A253665(n).

Cf. A328002 (row sums), A163590.

Programs

Maple

T := (n, k) -> 2^n*mul((j/2)^((-1)^(j - 1)), j = 1 .. k):
seq(seq(T(n, k), k=0..n), n=0..9);
# Recurrence:
Trow := proc(n) if n = 0 then return [1] fi; Trow(n - 1);
n^irem(n, 2) * (4/n)^irem(n + 1, 2) * %[n]; [op(2 * %%), %] end:
seq(print(Trow(n)), n = 0..9);

PARI

t(n, k) = 2^n * prod(j=1, k, ((j/2)^((-1)^(j - 1))))
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print upper 10 rows of the triangle as follows: */
trianglerows(10) \\ Felix Fröhlich, Apr 22 2021