A144644 Triangle in A144643 read by columns downwards.

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

Offset: 0

Author

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

Keywords

• nonn
• tabl

Comments

The Bell transform of the sequence "g(n) = 1 if n<4 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,    1;
  0, 1,  7,    6,     1;
  0, 0, 15,   25,    10,      1;
  0, 0, 25,   90,    65,     15,      1;
  0, 0, 35,  280,   350,    140,     21,     1;
  0, 0, 35,  770,  1645,   1050,    266,    28,     1;
  0, 0,  0, 1855,  6930,   6825,   2646,   462,    36,    1;
  0, 0,  0, 3675, 26425,  39795,  22575,  5880,   750,   45,  1;
  0, 0,  0, 5775, 90475, 211750, 172095, 63525, 11880, 1155, 55, 1;
		

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), A111246, A122848, A144643, A144645, A151509, A151511, A264428.

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;
A144644:= func< n,k | t(k,n) >;
[A144644(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 11 2023

Mathematica

With[{r=15}, Table[BellY[n, k, {1,1,1,1}], {n,0,r}, {k,0,n}]]//Flatten (* Jan Mangaldan, May 22 2016 *)

PARI

\\ Function bell_matrix is defined in A264428.
B = bell_matrix( n -> {if(n < 4, 1, 0)}, 9); for(n = 0, 9, printp(); for(k = 1, n, print1(B[n,k], " "))); \\ Peter Luschny, Apr 17 2019

Sage

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<4 else 0, 12) # Peter Luschny, Jan 19 2016

Formula

Bivariate e.g.f. A144644(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G4(x)), where G4(x) = Sum_{i=1..4} x^i/i! is the e.g.f. of column 1. - R. J. Mathar, May 28 2019

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

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

T(n, k) = A144645(n, n-k).

T(n, k) = t(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)