A174377 Triangle T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 3, read by rows.

1, 1, 1, 1, 6, 1, 1, 9, 9, 1, 1, 12, 108, 12, 1, 1, 15, 180, 180, 15, 1, 1, 18, 270, 3240, 270, 18, 1, 1, 21, 378, 5670, 5670, 378, 21, 1, 1, 24, 504, 9072, 136080, 9072, 504, 24, 1, 1, 27, 648, 13608, 244944, 244944, 13608, 648, 27, 1, 1, 30, 810, 19440, 408240, 7348320, 408240, 19440, 810, 30, 1

Offset

0,5

Comments

Row sums are: {1, 2, 8, 20, 134, 392, 3818, 12140, 155282, 518456, 8205362, ...}.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 3.

T(n, n-k) = T(n, k).

T(2*n, n) = A221954(n+1). - G. C. Greubel, Nov 28 2021

Example

Triangle begins as:
  1;
  1,  1;
  1,  6,   1;
  1,  9,   9,     1;
  1, 12, 108,    12,      1;
  1, 15, 180,   180,     15,       1;
  1, 18, 270,  3240,    270,      18,      1;
  1, 21, 378,  5670,   5670,     378,     21,     1;
  1, 24, 504,  9072, 136080,    9072,    504,    24,   1;
  1, 27, 648, 13608, 244944,  244944,  13608,   648,  27,  1;
  1, 30, 810, 19440, 408240, 7348320, 408240, 19440, 810, 30,  1;

Mathematica

T[n_, k_, q_]:= If[Floor[n/2]>=k, n!*q^k/(n-k)!, n!*q^(n-k)/k!];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Sage)
f=factorial
def T(n, k, q): return f(n)*q^k/f(n-k) if ((n//2)>k-1) else f(n)*q^(n-k)/f(k)
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 28 2021

Crossrefs

Cf. A159623 (q=1), A174376 (q=2), this sequence (q=3), A174378 (q=4).

Cf. A221954.

Sequence in context: A140284 A143087 A144470 * A176151 A204001 A363291

Adjacent sequences: A174374 A174375 A174376 * A174378 A174379 A174380

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 17 2010

Extensions

Edited by G. C. Greubel, Nov 28 2021

Status

approved