cp's OEIS Frontend

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.

%I A281901 #23 Nov 21 2024 05:18:21
%S A281901 1,2,31,18252,1495388159,34155922905682979,
%T A281901 350521520018942991464535019,2371013832433361706367594400829713564440,
%U A281901 14584126149704606223764458141727351569547933381159988406,107640669875812795238625627484701500354901860426640161278022882392148747562
%N 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.
%C A281901 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.
%H A281901 Alois P. Heinz, <a href="/A281901/b281901.txt">Table of n, a(n) for n = 0..26</a>
%H A281901 Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="https://arxiv.org/abs/1701.08394">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394, 2017.
%H A281901 Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="/A144416/a144416.txt">On-Line Appendix I to "Analysis of the gift exchange problem"</a>
%H A281901 Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="/A144416/a144416_1.txt">On-Line Appendix II to "Analysis of the gift exchange problem"</a>
%H A281901 David Applegate and N. J. A. Sloane, <a href="https://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009.
%F A281901 a(n) = A144510(n+1,n) = A144512(n,n).
%p A281901 with(combinat):
%p A281901 b:= proc(n, i, t) option remember; `if`(t*i<n, 0,
%p A281901       `if`(n=0, `if`(t=0, 1, 0), add(b(n-i*j, i-1, t-j)*
%p A281901        multinomial(n, n-i*j, i$j)/j!, j=0..min(t, n/i))))
%p A281901     end:
%p A281901 a:= n-> add(b(j, n+1, n), j=0..(n+1)*n):
%p A281901 seq(a(n), n=0..10);
%t A281901 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 *)
%Y A281901 Main diagonal of A144510, A144512.
%Y A281901 Cf. A001515, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.
%K A281901 nonn
%O A281901 0,2
%A A281901 _Alois P. Heinz_, Feb 01 2017