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%I #36 May 22 2019 21:03:55
%S 1,7,6427,216864652,60790021361170,79397199549271412737,
%T 350521520018942991464535019,4247805448772073978048752721163278,
%U 122022975450467092259059357046375920848764,7449370563518425038119522091529589590475534631830
%N Number of scenarios in the Gift Exchange Game when a gift can be stolen at most 6 times.
%C The result from the recurrence has been confirmed up to a(63) by using an optimized version of equation (23) in the Applegate-Sloane paper. - _Lars Blomberg_, Feb 01 2017
%H Lars Blomberg, <a href="/A281358/b281358.txt">Table of n, a(n) for n = 0..88</a>
%H Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1701.08394">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394 [math.CO], 2017.
%H 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>, giving Type D recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
%H 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>, giving Type C recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
%H David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009.
%p with(combinat):
%p b:= proc(n, i, t) option remember; `if`(t*i<n, 0,
%p `if`(n=0, `if`(t=0, 1, 0), add(b(n-i*j, i-1, t-j)*
%p multinomial(n, n-i*j, i$j)/j!, j=0..min(t, n/i))))
%p end:
%p a:= n-> add(b(k, 7, n), k=0..7*n):
%p seq(a(n), n=0..12); # _Alois P. Heinz_, Feb 01 2017
%t t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 7*n := t[n, k] = Sum[(1/j!)*Product[k - m, {m, 1, j}]*t[n - 1, k - j - 1], {j, 0, 6}]; t[_, _] = 0; a[n_] := Sum[t[n, k], {k, 0, 7*n}]; Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Feb 18 2017 *)
%o (PARI) {a(n) = sum(i=n, 7*n, i!*polcoef(sum(j=1, 7, x^j/j!)^n, i))/n!} \\ _Seiichi Manyama_, May 22 2019
%Y The gift scenarios sequences when a gift can be stolen at most s times, for s = 1..9, are A001515, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jan 25 2017
%E More terms from _Lars Blomberg_, Feb 01 2017
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