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A144509 a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., 5, for 0 <= k <= 5n.

%I A144509 #27 May 22 2019 21:04:09
%S A144509 1,5,456,405408,1495388159,15467641899285,361207016885536095,
%T A144509 16557834064546698285496,1350410785161120363519024709,
%U A144509 182141025850694258874753732988078,38395944834298393218465758049745918098,12093097322244029427838390643054170162054258,5485321312184901565806045962453632525844948020084
%N A144509 a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., 5, for 0 <= k <= 5n.
%C A144509 Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most 4 times. - _N. J. A. Sloane_, Jan 25 2017
%H A144509 Alois P. Heinz, <a href="/A144509/b144509.txt">Table of n, a(n) for n = 0..100</a>
%H A144509 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 A144509 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 A144509 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 A144509 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.
%t A144509 t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 5*n := t[n, k] = Sum[(1/j!)*Product[k - m, {m, 1, j}]*t[n - 1, k - j - 1], {j, 0, 4}]; t[_, _] = 0; a[n_] := Sum[t[n, k], {k, 0, 5*n}]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Feb 18 2017 *)
%o A144509 (PARI) {a(n) = sum(i=n, 5*n, i!*polcoef(sum(j=1, 5, x^j/j!)^n, i))/n!} \\ _Seiichi Manyama_, May 22 2019
%Y A144509 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 A144509 nonn
%O A144509 0,2
%A A144509 _David Applegate_ and _N. J. A. Sloane_, Dec 15 2008