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
KEYWORD
nonn,tabl
AUTHOR
David Applegate and N. J. A. Sloane, Jan 25 2009
STATUS
approved