A137268 Triangle T(n, k) = k! * (k+1)^(n-k), read by rows.
1, 2, 2, 4, 6, 6, 8, 18, 24, 24, 16, 54, 96, 120, 120, 32, 162, 384, 600, 720, 720, 64, 486, 1536, 3000, 4320, 5040, 5040, 128, 1458, 6144, 15000, 25920, 35280, 40320, 40320, 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880
Offset: 1
Examples
Triangle begins as:
1;
2, 2;
4, 6, 6;
8, 18, 24, 24;
16, 54, 96, 120, 120;
32, 162, 384, 600, 720, 720;
64, 486, 1536, 3000, 4320, 5040, 5040;
128, 1458, 6144, 15000, 25920, 35280, 40320, 40320;
256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 362880;
512, 13122, 98304, 375000, 933120, 1728720, 2580480, 3265920, 3628800, 3628800;
Links
- G. C. Greubel, Rows n = 1..50 of the triangle, flattened
- Fan Chung and R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194.
Programs
-
Magma
[Factorial(k)*(k+1)^(n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 28 2022
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Mathematica
T[n_, k_]:= k!*(k+1)^(n-k); Table[T[n, k], {n, 12}, {k, n}]//Flatten -
SageMath
def A137268(n,k): return factorial(k)*(k+1)^(n-k) flatten([[A137268(n,k) for k in range(1,n+1)] for n in range(14)]) # G. C. Greubel, Nov 28 2022
Formula
J(b, n) = (b+1)^(n-b)*b! if n > b, otherwise n! (notation of Chung and Graham).
From G. C. Greubel, Nov 28 2022: (Start)
T(n, k) = k! * (k+1)^(n-k).
T(n, n-2) = 2*A074143(n), n > 1.
T(2*n, n) = A152684(n).
T(2*n, n-1) = A061711(n).
T(2*n+1, n+1) = A066319(n). (End)
Extensions
Edited by G. C. Greubel, Nov 28 2022
Comments