A144385 -id:A144385 - cp's OEIS frontend

A151359 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3, 4, 5 or 6 (n >= 0, 0 <= k <= 6n).

0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 3, 7, 15, 31, 63, 119, 210, 336, 462, 462, 0, 0, 0, 1, 6, 25, 90, 301, 966, 2989, 8925, 25641, 70455, 183183, 441441, 966966, 1849848, 2858856, 2858856, 0, 0, 0, 0, 1, 10, 65, 350, 1701, 7770, 33985, 143605, 588511

Offset: 0

N. J. A. Sloane, May 14 2009

Row n has 6n+1 entries.

			Triangle begins:
[0, 1, 1, 1, 1, 1, 1]
[0, 0, 1, 3, 7, 15, 31, 63, 119, 210, 336, 462, 462]
[0, 0, 0, 1, 6, 25, 90, 301, 966, 2989, 8925, 25641, 70455, 183183, 441441, 966966, 1849848, 2858856, 2858856]
[0, 0, 0, 0, 1, 10, 65, 350, 1701, 7770, 33985, 143605, 588511, 2341339, 9032023, 33668635, 120681561, 413104692, 1337944608, 4046710668, 11216721516, 27756632904, 58555088592, 96197645544, 96197645544]
[0, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42525, 246400, 1370985, 7383376, 38657619, 197212015, 980839860, 4752728981, 22399494117, 102410296989, 452572985865, 1924000439361, 7820764020069, 30157961878044, 109184327692440, 365935843649376, 1113006758944080, 2982608000091720, 6696799094545560, 11423951396577720, 11423951396577720]
...

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 [math.CO], 2009.

This is one of a sequence of triangles: A144331, A144385, A144643, A151338, A151359, ...

See A151511, A151512 for other versions.

Unprotect[Power]; 0^0 = 1; a[n_ /; 1 <= n <= 6] = 1; a[] = 0; t[n, k_] := t[n, k] = If[k == 0, a[0]^n, Sum[Binomial[n-1, j-1] a[j] t[n-j, k-1], {j, 0, n-k+1}]]; T[n_, k_] := t[k, n+1]; Table[Table[T[n, k], {k, 0, 6(n+1)} ], {n, 0, 4}] // Flatten (* Jean-François Alcover, Jan 20 2016, using Peter Luschny's Bell transform *)

A144626 Tetrahedron of numbers T(i,j,k) = (i+2j+3k)!/(i!j!k!2^j6^k) read with entries in the order defined in A144625.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 4, 10, 10, 1, 6, 15, 15, 10, 60, 105, 70, 280, 280, 1, 10, 45, 105, 105, 20, 210, 840, 1260, 280, 2520, 6300, 2800, 15400, 15400, 1, 15, 105, 420, 945, 945, 35, 560, 3780, 12600, 17325, 840, 12600, 69300, 138600, 15400, 184800, 600600, 200200, 1401400, 1401400

Offset: 0

N. J. A. Sloane, Jan 18 2009, Jan 19 2009

The slice sums are given by A144416.

			The n-th slice of the tetrahedrom consists of the terms T(i,j,k) with i+j+k = n.
Slices 0,1,2,3,4,5 are:
....................1
------------------
....................1
...................1.1
------------------
....................10
...................4.10
..................1.3..3
------------------
...................280
..................70.280
................10.60.105
...............1..6.15..15
------------------
..................15400
................2800.15400
...............280.2520.6300
.............20..210.840.1260
............1..10..45..105.105
------------------
.................1401400
.............200200.1401400
.........15400.184800...600600
......840..12600..69300...138600
...35....560....3780....12600...17325
1......15....105....420......945.....945

Cf. A144416, A144625, A144385.

A144516 a(n) = (15n^2+45n-70)*binomial(n+4,6)/8.

Original entry on oeis.org

0, 0, 10, 175, 1225, 5565, 19425, 56595, 144375, 332475, 705705, 1401400, 2632630, 4718350, 8121750, 13498170, 21754050, 34118490, 52229100, 78233925, 114911335, 165809875, 235410175, 329311125, 454442625, 619307325, 834253875, 1111784310

Offset: 0

N. J. A. Sloane, Dec 17 2008

nonn

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

A diagonal of A144385.

f:=n->(15*n^2+45*n-70)*binomial(n+4,6)/8;

Table[((15n^2+45n-70)Binomial[n+4,6])/8,{n,0,30}] (* Harvey P. Dale, Dec 12 2018 *)

G.f.: 5x^2(2+17x+2x^2)/(1-x)^9. - R. J. Mathar, Jan 17 2009

cp's OEIS Frontend

A151359 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3, 4, 5 or 6 (n >= 0, 0 <= k <= 6n).

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A144626 Tetrahedron of numbers T(i,j,k) = (i+2j+3k)!/(i!j!k!2^j6^k) read with entries in the order defined in A144625.

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A144516 a(n) = (15n^2+45n-70)*binomial(n+4,6)/8.

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Formula

cp's OEIS Frontend

A151359 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3, 4, 5 or 6 (n >= 0, 0 <= k <= 6n).

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Mathematica

A144626 Tetrahedron of numbers T(i,j,k) = (i+2*j+3*k)!/(i!*j!*k!*2^j*6^k) read with entries in the order defined in A144625.

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Author

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Examples

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A144516 a(n) = (15*n^2+45*n-70)*binomial(n+4,6)/8.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A144626 Tetrahedron of numbers T(i,j,k) = (i+2j+3k)!/(i!j!k!2^j6^k) read with entries in the order defined in A144625.

A144516 a(n) = (15n^2+45n-70)*binomial(n+4,6)/8.