A174378 Triangle T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 4, read by rows.

1, 1, 1, 1, 8, 1, 1, 12, 12, 1, 1, 16, 192, 16, 1, 1, 20, 320, 320, 20, 1, 1, 24, 480, 7680, 480, 24, 1, 1, 28, 672, 13440, 13440, 672, 28, 1, 1, 32, 896, 21504, 430080, 21504, 896, 32, 1, 1, 36, 1152, 32256, 774144, 774144, 32256, 1152, 36, 1, 1, 40, 1440, 46080, 1290240, 30965760, 1290240, 46080, 1440, 40, 1

Offset

0,5

Comments

Row sums are: {1, 2, 10, 26, 226, 682, 8690, 28282, 474946, 1615178, ...}.

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 = 2.

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

T(2*n, n) = A052734(n+1). - G. C. Greubel, Nov 28 2021

Example

Triangle begins as:
  1;
  1,  1;
  1,  8,    1;
  1, 12,   12,     1;
  1, 16,  192,    16,       1;
  1, 20,  320,   320,      20,        1;
  1, 24,  480,  7680,     480,       24,       1;
  1, 28,  672, 13440,   13440,      672,      28,     1;
  1, 32,  896, 21504,  430080,    21504,     896,    32,    1;
  1, 36, 1152, 32256,  774144,   774144,   32256,  1152,   36,  1;
  1, 40, 1440, 46080, 1290240, 30965760, 1290240, 46080, 1440, 40,  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, 4], {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, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 28 2021

Crossrefs

Cf. A159623 (q=1), A174376 (q=2), A174377 (q=3), this sequence (q=4).

Cf. A052734.

Sequence in context: A173742 A146881 A174301 * A131067 A157170 A143679

Adjacent sequences: A174375 A174376 A174377 * A174379 A174380 A174381

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 17 2010

Extensions

Edited by G. C. Greubel, Nov 28 2021

Status

approved