A144508 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.

1, 4, 121, 18252, 7958726, 7528988476, 13130817809439, 38001495237579931, 169490425291053577442, 1102725620990181693266071, 10030550674270068548738783696, 123317200510025161580777179001154, 1993320784474917266370637900936051186, 41401645296339316791633672053851083955147

Offset: 0

Author

David Applegate and N. J. A. Sloane, Dec 15 2008

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix I to "Analysis of the gift exchange problem", giving Type D recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix II to "Analysis of the gift exchange problem", giving Type C recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
• David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Crossrefs

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.

Programs

Mathematica

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 *)

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