login
a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, 3 or 4, for 0 <= k <= 4n.
14

%I #26 May 22 2019 21:04:15

%S 1,4,121,18252,7958726,7528988476,13130817809439,38001495237579931,

%T 169490425291053577442,1102725620990181693266071,

%U 10030550674270068548738783696,123317200510025161580777179001154,1993320784474917266370637900936051186,41401645296339316791633672053851083955147

%N a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, 3 or 4, for 0 <= k <= 4n.

%C Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most 3 times. - _N. J. A. Sloane_, Jan 25 2017

%H Alois P. Heinz, <a href="/A144508/b144508.txt">Table of n, a(n) for n = 0..100</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.

%t t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 4*n := t[n, k] = t[n - 1, k - 1] + (k - 1)*t[n - 1, k - 2] + (1/2)*(k - 1)*(k - 2)*t[n - 1, k - 3] + (1/6)*(k - 1)*(k - 2)*(k - 3)*t[n - 1, k - 4]; t[_, _] = 0; a[n_] := Sum[t[n, k], {k, 0, 4*n}]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Feb 18 2017 *)

%o (PARI) {a(n) = sum(i=n, 4*n, i!*polcoef(sum(j=1, 4, 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 _David Applegate_ and _N. J. A. Sloane_, Dec 15 2008