A180683 T(n,k) is the sum of the path counts in the (right-aligned Ferrers plots of) the partitions of n in exactly k parts.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 6, 6, 3, 1, 1, 1, 6, 9, 6, 3, 1, 1, 1, 10, 14, 13, 6, 3, 1, 1, 1, 10, 24, 18, 13, 6, 3, 1, 1, 1, 15, 31, 35, 23, 13, 6, 3, 1, 1, 1, 15, 45, 51, 40, 23, 13, 6, 3, 1, 1, 1, 21, 64, 85, 65, 46, 23, 13, 6, 3, 1, 1, 1, 21, 82, 118, 111, 71, 46, 23, 13, 6, 3

Offset

1,8

Comments

Reverse of last row converges to middle value of odd rows: 1, 1, 3, 6, 13, 23, 46, 78, 143, 240, 414, 673, 1127, 1788, 2885, 4514, 7096,10885, 16784, 25338, 38347, 57147, 85094, 125157, ...

Contribution from Robert G. Wilson v, Sep 25 2010: (Start)

\k..1....2....3....4....5....6....7....8....9...10...11...12...13...14...15...16...17...18

n\

..

.1..1

.2..1....1

.3..1....1....1

.4..1....3....1....1

.5..1....3....3....1....1

.6..1....6....6....3....1....1

.7..1....6....9....6....3....1....1

.8..1...10...14...13....6....3....1....1

.9..1...10...24...18...13....6....3....1....1

10..1...15...31...35...23...13....6....3....1....1

11..1...15...45...51...40...23...13....6....3....1....1

12..1...21...64...85...65...46...23...13....6....3....1....1

13..1...21...82..118..111...71...46...23...13....6....3....1....1

14..1...28..107..181..171..128...78...46...23...13....6....3....1....1

15..1...28..144..244..268..203..135...78...46...23...13....6....3....1....1

16..1...36..175..362..393..334..223..143...78...46...23...13....6....3....1....1

17..1...36..221..470..590..503..372..231..143...78...46...23...13....6....3....1....1

18..1...45..279..654..844..800..582..395..240..143...78...46...23...13....6....3....1....1

... (End)

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..5050.

Mathematica

pathcount[p_] := Block[{ferr = (0*Range[#1] &) /@ p}, Last[ Fold[ Rest[ FoldList[ Plus, 0, Drop[#1, Length[#1] - Length[#2]] + #2]] &, 1 + First[ferr], Rest[ferr]]]]; t[n_, k_] := Plus @@ pathcount /@ IntegerPartitions[n, {k}]; Table[ t[n, k], {n, 13}, {k, n}] // Flatten

Crossrefs

Cf. A000012, A008805. [From Robert G. Wilson v, Sep 25 2010]

Sequence in context: A295931 A295920 A176187 * A365331 A214635 A166030

Adjacent sequences: A180680 A180681 A180682 * A180684 A180685 A180686

Keyword

nonn,tabl

Author

Wouter Meeussen, Sep 16 2010

Status

approved