A193282 -id:A193282 - cp's OEIS frontend

A349269 Triangle read by rows, T(n, k) = (n - k)! * k! / floor(k / 2)! ^ 2.

1, 1, 1, 2, 1, 2, 6, 2, 2, 6, 24, 6, 4, 6, 6, 120, 24, 12, 12, 6, 30, 720, 120, 48, 36, 12, 30, 20, 5040, 720, 240, 144, 36, 60, 20, 140, 40320, 5040, 1440, 720, 144, 180, 40, 140, 70, 362880, 40320, 10080, 4320, 720, 720, 120, 280, 70, 630

Offset: 0

Peter Luschny, Nov 13 2021

Interpolates between the factorial numbers (A000142) and the swinging factorial numbers (A056040).

The identity T(n, 0) = T(n, n)*T(floor(n/2), 0)^2 was investigated as a basis for an efficient implementation of the computation of the factorial numbers (see link).

			[0]      1;
[1]      1,     1;
[2]      2,     1,     2;
[3]      6,     2,     2,    6;
[4]     24,     6,     4,    6,   6;
[5]    120,    24,    12,   12,   6,  30;
[6]    720,   120,    48,   36,  12,  30,  20;
[7]   5040,   720,   240,  144,  36,  60,  20, 140;
[8]  40320,  5040,  1440,  720, 144, 180,  40, 140, 70;
[9] 362880, 40320, 10080, 4320, 720, 720, 120, 280, 70, 630;

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Cf. A349270 (row sums), A193282 (central coeffs.), A000142, A056040, A180064.

T := (n, k) -> (n - k)!*k! / iquo(k,2)! ^ 2:
seq(seq(T(n, k), k = 0..n), n = 0..9);

T(n, k) divides T(n, 0) for 0 <= k <= n.

Product_{k=0..n} T(n, k) is a square.

A356636 Triangle read by rows. T(n, k) = binomial(n, k) * n!^2 / floor(n/2)!^2.

Original entry on oeis.org

1, 1, 1, 4, 8, 4, 36, 108, 108, 36, 144, 576, 864, 576, 144, 3600, 18000, 36000, 36000, 18000, 3600, 14400, 86400, 216000, 288000, 216000, 86400, 14400, 705600, 4939200, 14817600, 24696000, 24696000, 14817600, 4939200, 705600

Offset: 0

Peter Luschny, Aug 19 2022

			Triangle T(n, k) starts:
[0]     1;
[1]     1,     1;
[2]     4,     8,      4;
[3]    36,   108,    108,     36;
[4]   144,   576,    864,    576,    144;
[5]  3600, 18000,  36000,  36000,  18000,  3600;
[6] 14400, 86400, 216000, 288000, 216000, 86400, 14400;

Cf. A056040, A193282, A253666.

A356636 := (n, k) -> binomial(n, k) * (n! / iquo(n, 2)!) ^ 2:
for n from 0 to 9 do seq(A356636(n, k), k = 0..n) od;

T(n, 0) = T(n, n) = A193282(n).

Sum_{k=0..n} T(n, k) = 2^n * A193282(n).

cp's OEIS Frontend

A349269 Triangle read by rows, T(n, k) = (n - k)! * k! / floor(k / 2)! ^ 2.

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