A281901 Number of scenarios in the Gift Exchange Game with n players and n wrapped gifts when a gift can be stolen at most n times.

1, 2, 31, 18252, 1495388159, 34155922905682979, 350521520018942991464535019, 2371013832433361706367594400829713564440, 14584126149704606223764458141727351569547933381159988406, 107640669875812795238625627484701500354901860426640161278022882392148747562

Offset

0,2

Comments

Also total number of partitions of [k] into exactly n nonempty blocks, each of size at most n+1, for any k in the range n <= k <= n^2+n.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..26

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix I to "Analysis of the gift exchange problem"

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix II to "Analysis of the gift exchange problem"

David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Formula

a(n) = A144510(n+1,n) = A144512(n,n).

Maple

with(combinat):
b:= proc(n, i, t) option remember; `if`(t*i<n, 0,
      `if`(n=0, `if`(t=0, 1, 0), add(b(n-i*j, i-1, t-j)*
       multinomial(n, n-i*j, i$j)/j!, j=0..min(t, n/i))))
    end:
a:= n-> add(b(j, n+1, n), j=0..(n+1)*n):
seq(a(n), n=0..10);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, t_] := b[n, i, t] = If[t*i<n, 0, If[n==0, If[t==0, 1, 0], Sum[b[n-i*j, i-1, t-j]* multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!, {j, 0, Min[t, n/i]}]]]; a[n_] := Sum[b[j, n+1, n], {j, 0, (n+1)*n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Mar 13 2017, translated from Maple *)

Crossrefs

Main diagonal of A144510, A144512.

Cf. A001515, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.

Sequence in context: A004072 A113030 A247873 * A357176 A018802 A166487

Adjacent sequences: A281898 A281899 A281900 * A281902 A281903 A281904

Keyword

nonn

Author

Alois P. Heinz, Feb 01 2017

Status

approved