A147984 -id:A147984

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A144512

Array read by upwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., k+1, for 0 <= k <= (k+1)*n.

+10
6

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 31, 37, 1, 1, 5, 121, 842, 266, 1, 1, 6, 456, 18252, 45296, 2431, 1, 1, 7, 1709, 405408, 7958726, 4061871, 27007, 1, 1, 8, 6427, 9268549, 1495388159, 7528988476, 546809243, 353522, 1, 1, 9, 24301, 216864652, 295887993624, 15467641899285

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..50.

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.

David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

EXAMPLE

Array begins:

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, ...

1, 3, 31, 842, 45296, 4061871, 546809243, 103123135501, ...

1, 4, 121, 18252, 7958726, 7528988476, 13130817809439, ...

1, 5, 456, 405408, 1495388159, 15467641899285, 361207016885536095, ...

1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ...

...

MAPLE

b := proc(n, i, k) local r;

option remember;

if n = i then 1;

elif i < n then 0;

elif n < 1 then 0;

else add( binomial(i-1, r)*b(n-1, i-1-r, k), r=0..k);

end if;

end proc;

T:=proc(n, k); add(b(n, i, k), i=0..(k+1)*n); end proc;

MATHEMATICA

multinomial[n_, k_List] := n!/Times @@ (k!); t[n_, k_] := Module[{i, ik}, ik = Array[i, k]; 1/k!* Sum[multinomial[Total[ik], ik], Evaluate[Sequence @@ Thread[{ik, 1, n}]]]]; Table[t[n-k, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

CROSSREFS

See A144510 for Maple code.

Rows include A001515, A144416, A144508, A144509, A149187.

Columns include A048775, A144511, A144662, A147984.

Transpose of array in A144510.

Main diagonal gives A281901.

KEYWORD

nonn,tabl

AUTHOR

David Applegate and N. J. A. Sloane, Dec 15 2008, Dec 21 2008

STATUS

approved

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