A159623 Triangle read by rows: T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 1.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 12, 4, 1, 1, 5, 20, 20, 5, 1, 1, 6, 30, 120, 30, 6, 1, 1, 7, 42, 210, 210, 42, 7, 1, 1, 8, 56, 336, 1680, 336, 56, 8, 1, 1, 9, 72, 504, 3024, 3024, 504, 72, 9, 1, 1, 10, 90, 720, 5040, 30240, 5040, 720, 90, 10, 1

Offset: 0

Author

Roger L. Bagula, Apr 17 2009

Keywords

• nonn
• tabl
• easy

Comments

The row sums are: {1, 2, 4, 8, 22, 52, 194, 520, 2482, 7220, 41962,...}.

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  4, 12,   4,    1;
  1,  5, 20,  20,    5,     1;
  1,  6, 30, 120,   30,     6,    1;
  1,  7, 42, 210,  210,    42,    7,   1;
  1,  8, 56, 336, 1680,   336,   56,   8,  1;
  1,  9, 72, 504, 3024,  3024,  504,  72,  9,  1;
  1, 10, 90, 720, 5040, 30240, 5040, 720, 90, 10,  1;
		

Links

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

Crossrefs

Cf. this sequence (q=1), A174376 (q=2), A174377 (q=3), A174378 (q=4).

Programs

Mathematica

T[n_, k_, q_]:= If[Floor[n/2]>=k, n!*q^k/(n-k)!, n!*q^(n-k)/k!];
Table[T[n, k, 1], {n,0,12}, {k,0,n}]//Flatten

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,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 28 2021

Formula

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

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

T(2*n, n) = A001813(n). - G. C. Greubel, Nov 28 2021

Extensions

Edited by N. J. A. Sloane, Apr 17 2009