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

0,9

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

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.

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)

Example

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;

Mathematica

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

Prog

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

Crossrefs

Cf. A001681 (row sums), A111246, A122848, A144643, A144645, A151509, A151511, A264428.

Sequence in context: A213060 A272008 A054024 * A151509 A264434 A151511

Adjacent sequences: A144641 A144642 A144643 * A144645 A144646 A144647

Keyword

nonn,tabl

Author

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

Status

approved