A144626 -id:A144626 - cp's OEIS frontend

A144416 a(n) is the total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2 or 3, for 0 <= k <= 3n.

1, 3, 31, 842, 45296, 4061871, 546809243, 103123135501, 25942945219747, 8394104851717686, 3395846808758759686, 1679398297627675722593, 996789456118195908366641, 699283226713639676370419067, 572385833490097906671186099971, 540635257271794961275858251107746, 583630397618757664934692641037584628

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008, Dec 17 2008

nonn

Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most twice. - N. J. A. Sloane, Jan 25 2017

			a(0) = 1;
a(1) = 3: {1} {12} {123}
a(2) = 31: {1,2} {1,23} {2,13} {3,12} {1,234} {2,134} {3,124} {4,123}
{12,34} {13,24} {14,23} {12,345} {13,245} {14,235} {15,234} {23,145} {24,135}
{25,134} {34,125} {35,124} {45,123} {123,456} {124,356} {125,346} {126,345}
{134,256} {135,246} {136,245} {145,236} {146,235} {156,234}.

David Applegate and N. J. A. Sloane, 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.

Row sums of A144385. Slice sums of A144626.

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.

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 3*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]; t[, ] = 0; a[n_] := Sum[t[n, k], {k, 0, 3*n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 18 2017 *)

{a(n) = sum(i=n, 3*n, i!*polcoef(sum(j=1, 3, x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 22 2019

a(n) = Sum_{ b,c >= 0, b+c <= n } (n+b+2c)!/ ((n-b-c)! b! c! 2^b 6^c).

The sum is dominated by the b=0, c=n term, so a(n) ~ constant*(3*n)!/(n!*6^n).

A182631 Tetrahedron in which T(i,j,k) is the next number congruent to k mod j of the level j in the slice i, with 0 <= i, 1 <= j, 0 <= k < j.

Original entry on oeis.org

0, 1, 0, 1, 2, 2, 3, 0, 1, 2, 3, 4, 5, 3, 4, 5, 0, 1, 2, 3, 4, 6, 7, 6, 7, 8, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 8, 9, 9, 10, 11, 8, 9, 10, 11, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 12, 13, 14, 15, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6

Offset: 0

Omar E. Pol, Dec 06 2010

			Tetrahedron begins:
For i=0, j=1, k=0, T(0,1,0)=0 represents the first vertex of the tetrahedron.
For i=1, slice 1 lists the terms
  1;
  0, 1.
For i=2, slice 2 lists the terms
  2;
  2, 3;
  0, 1, 2.
For i=3, slice 3 lists the terms
  3;
  4, 5;
  3, 4, 5;
  0, 1, 2, 3.
For i=4, slice 4 lists the terms
  4;
  6, 7;
  6, 7, 8;
  4, 5, 6, 7;
  0, 1, 2, 3, 4.
For i=5, slice 5 lists the terms
  5;
  8,  9;
  9, 10, 11;
  8,  9, 10, 11;
  5,  6,  7,  8,  9;
  0,  1,  2,  3,  4,  5.
And so on.
If the sequence is written as a triangle, it begins:
  0,
  1, 0, 1,
  2, 2, 3, 0, 1, 2,
  3, 4, 5, 3, 4, 5, 0, 1, 2, 3,
  4, 6, 7, 6, 7, 8, 4, 5, 6, 7, 0, 1, 2, 3, 4;
  ...

Cf. A144626.
Level j=1 column k=0 of tetrahedron = column 1 of triangle gives A001477.
Level j=2 column k=0 of tetrahedron = column 2 of triangle gives A005843.
Level j=2 column k=1 of tetrahedron = column 3 of triangle gives A005408.
Level j=3 column k=0 of tetrahedron = column 4 of triangle gives A008585.
Level j=3 column k=1 of tetrahedron = column 5 of triangle gives A016777.
Level j=3 column k=2 of tetrahedron = column 6 of triangle gives A016789.
And so on.

cp's OEIS Frontend

A144416 a(n) is the total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2 or 3, for 0 <= k <= 3n.

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