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A144643 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3 or 4 (n >= 0, 0 <= k <= 4n). 6
1, 0, 1, 1, 1, 1, 0, 0, 1, 3, 7, 15, 25, 35, 35, 0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775, 0, 0, 0, 0, 1, 10, 65, 350, 1645, 6930, 26425, 90475, 275275, 725725, 1576575, 2627625, 2627625, 0, 0, 0, 0, 0, 1, 15, 140, 1050, 6825, 39795 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

LINKS

Table of n, a(n) for n=0..55.

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)

FORMULA

For recurrence see Maple code.

EXAMPLE

Triangle begins:

1,

0, 1, 1, 1, 1,

0, 0, 1, 3, 7, 15, 25, 35, 35,

0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775,

0, 0, 0, 0, 1, 10, 65, 350, 1645, 6930, 26425, 90475, 275275, 725725, 1576575, 2627625, 2627625

0, 0, 0, 0, 0, 1, 15, 140, 1050, 6825, 39795, 211750, 1033725, 4629625, 18918900, 69719650, 227727500, 640264625, 1474097625, 2546168625, 2546168625

MAPLE

T := proc(n, k) option remember;

if n = k then 1;

elif k < n then 0;

elif n < 1 then 0;

else 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);

end if;

end proc;

MATHEMATICA

T[n_, k_] := T[n, k] = Which[n == k, 1, k<n, 0, n<1, 0, True, 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]]; Table[T[n, k], {n, 0, 5}, {k, 0, 4n}] // Flatten (* Jean-Fran├žois Alcover, Mar 20 2014, after Maple *)

CROSSREFS

Row sums give A144508. See A144644 and A144645 for other versions.

Cf. A144299, A144385.

Sequence in context: A289828 A226471 A175510 * A034757 A328688 A291651

Adjacent sequences:  A144640 A144641 A144642 * A144644 A144645 A144646

KEYWORD

nonn,tabf

AUTHOR

David Applegate and N. J. A. Sloane, Jan 25 2009

STATUS

approved

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Last modified May 25 10:57 EDT 2020. Contains 334592 sequences. (Running on oeis4.)