A144502 Square array read by antidiagonals upwards: T(n,k) is the number of scenarios for the gift exchange problem in which each gift can be stolen at most once, when there are n gifts in the pool and k gifts (not yet frozen) in peoples' hands.

1, 1, 1, 2, 2, 1, 7, 7, 5, 1, 37, 37, 30, 16, 1, 266, 266, 229, 155, 65, 1, 2431, 2431, 2165, 1633, 946, 326, 1, 27007, 27007, 24576, 19714, 13219, 6687, 1957, 1, 353522, 353522, 326515, 272501, 198773, 119917, 53822, 13700, 1, 5329837, 5329837, 4976315, 4269271, 3289726, 2199722, 1205857, 486355, 109601, 1

Offset: 0

Author

David Applegate and N. J. A. Sloane, Dec 13 2008

Keywords

• nonn
• tabl

Examples

			The array, A(n,k), begins:
    1,    1,     1,      1,       1,        1, ...
    1,    2,     5,     16,      65,      326, ...
    2,    7,    30,    155,     946,     6687, ...
    7,   37,   229,   1633,   13219,   119917, ...
   37,  266,  2165,  19714,  198773,  2199722, ...
  266, 2431, 24576, 272501, 3289726, 42965211, ...
  ...
Antidiagonal triangle, T(n,k), begins as:
      1;
      1,     1;
      2,     2,     1;
      7,     7,     5,     1;
     37,    37,    30,    16,     1;
    266,   266,   229,   155,    65,    1;
   2431,  2431,  2165,  1633,   946,  326,    1;
  27007, 27007, 24576, 19714, 13219, 6687, 1957,   1;
		

Links

• G. C. Greubel, Antidiagonals n = 0..50, 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

Rows include A000522, A144495, A144496, A144497.

Columns include A144301, A001515, A144498, A144499, A144500.

Main diagonal is A144501.

Antidiagonal sums give A144503.

Programs

Magma

A144301:= func< n | (&+[ Binomial(n+k-1,2*k)*Factorial(2*k)/( Factorial(k)*2^k): k in [0..n]]) >;
function A(n,k)
  if n eq 0 then return 1;
  elif k eq 0 then return A144301(n);
  elif k eq 1 then return A144301(n+1);
  else return A(n-1,k+1) + k*A(n,k-1);
  end if;
end function;
A144502:= func< n,k | A(n-k, k) >;
[A144502(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2023

Maple

B:=proc(p,r) option remember;
if p=0 then RETURN(1); fi;
if r=0 then RETURN(B(p-1,1)); fi;
B(p-1,r+1)+r*B(p,r-1); end;
seq(seq(B(d-k, k), k=0..d), d=0..9);

Mathematica

t[0, ]= 1; t[n, 0]:= t[n, 0]= t[n-1, 1];
t[n_, k_]:= t[n, k]= t[n-1, k+1] + k*t[n, k-1];
Table[t[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jan 14 2014, after Maple *)

SageMath

def A144301(n): return 1 if n<2 else (2*n-3)*A144301(n-1)+A144301(n-2)
@CachedFunction
def A(n,k):
    if n==0: return 1
    elif k==0: return A144301(n)
    elif k==1: return A144301(n+1)
    else: return A(n-1,k+1) + k*A(n,k-1)
def A144502(n,k): return A(n-k,k)
flatten([[A144502(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 29 2023

Formula

Let A_n(x) be the e.g.f. for row n. Then A_0(x) = exp(x) and for n >= 1, A_n(x) = (d/dx)A_{n-1}(x)/(1-x).

For n >= 1, the rows A_{n}(x) = P_{n}(x)*exp(x)/(1-x)^(2*n), where P_{n}(x) = (1-x)*(d/dx)( P_{n-1}(x) ) + (2*n-x)*P_{n-1}(x) and P_{0}(x) = 1. - G. C. Greubel, Oct 08 2023

Extensions

6 more terms from Michel Marcus, Feb 01 2023