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Triangle T(n, k) = k! * (k+1)^(n-k), read by rows.
3

%I #17 Nov 29 2022 01:33:24

%S 1,2,2,4,6,6,8,18,24,24,16,54,96,120,120,32,162,384,600,720,720,64,

%T 486,1536,3000,4320,5040,5040,128,1458,6144,15000,25920,35280,40320,

%U 40320,256,4374,24576,75000,155520,246960,322560,362880,362880

%N Triangle T(n, k) = k! * (k+1)^(n-k), read by rows.

%C Essentially the same as A104001.

%H G. C. Greubel, <a href="/A137268/b137268.txt">Rows n = 1..50 of the triangle, flattened</a>

%H Fan Chung and R. L. Graham, <a href="http://www.jstor.org/stable/27642443">Primitive juggling sequences</a>, Am. Math. Monthly 115 (3) (2008) 185-194.

%F J(b, n) = (b+1)^(n-b)*b! if n > b, otherwise n! (notation of Chung and Graham).

%F From _G. C. Greubel_, Nov 28 2022: (Start)

%F T(n, k) = k! * (k+1)^(n-k).

%F T(n, n-2) = 2*A074143(n), n > 1.

%F T(2*n, n) = A152684(n).

%F T(2*n, n-1) = A061711(n).

%F T(2*n+1, n+1) = A066319(n). (End)

%e Triangle begins as:

%e 1;

%e 2, 2;

%e 4, 6, 6;

%e 8, 18, 24, 24;

%e 16, 54, 96, 120, 120;

%e 32, 162, 384, 600, 720, 720;

%e 64, 486, 1536, 3000, 4320, 5040, 5040;

%e 128, 1458, 6144, 15000, 25920, 35280, 40320, 40320;

%e 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880;

%e 512, 13122, 98304, 375000, 933120, 1728720, 2580480, 3265920, 3628800, 3628800;

%t T[n_, k_]:= k!*(k+1)^(n-k);

%t Table[T[n, k], {n, 12}, {k, n}]//Flatten

%o (Magma) [Factorial(k)*(k+1)^(n-k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 28 2022

%o (SageMath)

%o def A137268(n,k): return factorial(k)*(k+1)^(n-k)

%o flatten([[A137268(n,k) for k in range(1,n+1)] for n in range(14)]) # _G. C. Greubel_, Nov 28 2022

%Y Cf. A061711, A066319, A074143, A104001, A152684.

%K nonn,tabl,easy

%O 1,2

%A _Roger L. Bagula_, Mar 12 2008

%E Edited by _G. C. Greubel_, Nov 28 2022