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

1,2

Comments

Essentially the same as A104001.

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.

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)

Example

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;

Mathematica

T[n_, k_]:= k!*(k+1)^(n-k);
Table[T[n, k], {n, 12}, {k, n}]//Flatten

Prog

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

Crossrefs

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

Sequence in context: A309075 A039731 A005341 * A008130 A055388 A065457

Adjacent sequences: A137265 A137266 A137267 * A137269 A137270 A137271

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 12 2008

Extensions

Edited by G. C. Greubel, Nov 28 2022

Status

approved