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A137268 Triangle T(n, k) = k! * (k+1)^(n-k), read by rows.

%I A137268 #17 Nov 29 2022 01:33:24
%S A137268 1,2,2,4,6,6,8,18,24,24,16,54,96,120,120,32,162,384,600,720,720,64,
%T A137268 486,1536,3000,4320,5040,5040,128,1458,6144,15000,25920,35280,40320,
%U A137268 40320,256,4374,24576,75000,155520,246960,322560,362880,362880
%N A137268 Triangle T(n, k) = k! * (k+1)^(n-k), read by rows.
%C A137268 Essentially the same as A104001.
%H A137268 G. C. Greubel, <a href="/A137268/b137268.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A137268 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 A137268 J(b, n) = (b+1)^(n-b)*b! if n > b, otherwise n! (notation of Chung and Graham).
%F A137268 From _G. C. Greubel_, Nov 28 2022: (Start)
%F A137268 T(n, k) = k! * (k+1)^(n-k).
%F A137268 T(n, n-2) = 2*A074143(n), n > 1.
%F A137268 T(2*n, n) = A152684(n).
%F A137268 T(2*n, n-1) = A061711(n).
%F A137268 T(2*n+1, n+1) = A066319(n). (End)
%e A137268 Triangle begins as:
%e A137268     1;
%e A137268     2,     2;
%e A137268     4,     6,     6;
%e A137268     8,    18,    24,     24;
%e A137268    16,    54,    96,    120,    120;
%e A137268    32,   162,   384,    600,    720,     720;
%e A137268    64,   486,  1536,   3000,   4320,    5040,    5040;
%e A137268   128,  1458,  6144,  15000,  25920,   35280,   40320,   40320;
%e A137268   256,  4374, 24576,  75000, 155520,  246960,  322560,  362880,  362880;
%e A137268   512, 13122, 98304, 375000, 933120, 1728720, 2580480, 3265920, 3628800, 3628800;
%t A137268 T[n_, k_]:= k!*(k+1)^(n-k);
%t A137268 Table[T[n, k], {n, 12}, {k, n}]//Flatten
%o A137268 (Magma) [Factorial(k)*(k+1)^(n-k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 28 2022
%o A137268 (SageMath)
%o A137268 def A137268(n,k): return factorial(k)*(k+1)^(n-k)
%o A137268 flatten([[A137268(n,k) for k in range(1,n+1)] for n in range(14)]) # _G. C. Greubel_, Nov 28 2022
%Y A137268 Cf. A061711, A066319, A074143, A104001, A152684.
%K A137268 nonn,tabl,easy
%O A137268 1,2
%A A137268 _Roger L. Bagula_, Mar 12 2008
%E A137268 Edited by _G. C. Greubel_, Nov 28 2022