A144645 Triangle in A144643 read upwards by columns.

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 0, 0, 1, 15, 65, 90, 25, 0, 0, 1, 21, 140, 350, 280, 35, 0, 0, 1, 28, 266, 1050, 1645, 770, 35, 0, 0, 1, 36, 462, 2646, 6825, 6930, 1855, 0, 0, 0, 1, 45, 750, 5880, 22575, 39795, 26425, 3675, 0, 0, 0

Offset: 0

Author

David Applegate and N. J. A. Sloane, Jan 25 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,  0;
  1,  1,   0;
  1,  3,   1,    0;
  1,  6,   7,    1,     0;
  1, 10,  25,   15,     0,     0;
  1, 15,  65,   90,    25,     0,     0;
  1, 21, 140,  350,   280,    35,     0,    0;
  1, 28, 266, 1050,  1645,   770,    35,    0,   0;
  1, 36, 462, 2646,  6825,  6930,  1855,    0,   0,   0;
  1, 45, 750, 5880, 22575, 39795, 26425, 3675,   0,   0,   0;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
• David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Crossrefs

Cf. A001681 (row sums), A144643, A144644.

Programs

Magma

function t(n,k)
  if k eq n then return 1;
  elif k le n-1 or n le 0 then return 0;
  else return (&+[Binomial(k-1,j)*t(n-1,k-j-1): j in [0..3]]);
  end if;
end function;
A144645:= func< n,k | t(n-k,n) >;
[A144645(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 11 2023

Mathematica

Table[BellY[n, n-k, {1,1,1,1}], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 11 2023; based on A144644 *)

SageMath

@CachedFunction
def t(n,k):
    if (k==n): return 1
    elif (kA144645(n,k): return t(n-k,n)
flatten([[A144645(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 11 2023

Formula

From G. C. Greubel, Oct 11 2023: (Start)

T(n, k) = A144643(n-k, n).

T(n, k) = A144644(n, n-k).

T(n, k) = t(n-k, n), where t(n, k) = Sum_{j=0..3} binomial(k-1, j) * t(n-1, k-j-1), with t(n,n) = 1, t(n,k) = 0 if n < 1 or n > k.

Sum_{k=0..n} T(n, k) = A001681(n). (End)