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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. 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-Fran├žois 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.

Sequence in context: A341014 A145085 A228904 * A159314 A135701 A051467

Adjacent sequences:  A144509 A144510 A144511 * A144513 A144514 A144515

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

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Last modified September 24 22:46 EDT 2021. Contains 347651 sequences. (Running on oeis4.)