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%I #11 Aug 28 2013 10:16:32
%S 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,
%T 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,
%U 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
%N 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.
%C 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, ...
%C Contribution from _Robert G. Wilson v_, Sep 25 2010: (Start)
%C \k..1....2....3....4....5....6....7....8....9...10...11...12...13...14...15...16...17...18
%C n\
%C ..
%C .1..1
%C .2..1....1
%C .3..1....1....1
%C .4..1....3....1....1
%C .5..1....3....3....1....1
%C .6..1....6....6....3....1....1
%C .7..1....6....9....6....3....1....1
%C .8..1...10...14...13....6....3....1....1
%C .9..1...10...24...18...13....6....3....1....1
%C 10..1...15...31...35...23...13....6....3....1....1
%C 11..1...15...45...51...40...23...13....6....3....1....1
%C 12..1...21...64...85...65...46...23...13....6....3....1....1
%C 13..1...21...82..118..111...71...46...23...13....6....3....1....1
%C 14..1...28..107..181..171..128...78...46...23...13....6....3....1....1
%C 15..1...28..144..244..268..203..135...78...46...23...13....6....3....1....1
%C 16..1...36..175..362..393..334..223..143...78...46...23...13....6....3....1....1
%C 17..1...36..221..470..590..503..372..231..143...78...46...23...13....6....3....1....1
%C 18..1...45..279..654..844..800..582..395..240..143...78...46...23...13....6....3....1....1
%C ... (End)
%H Robert G. Wilson v, <a href="/A180683/b180683.txt">Table of n, a(n) for n = 1..5050</a>.
%t 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
%Y Cf. A000012, A008805. [From _Robert G. Wilson v_, Sep 25 2010]
%K nonn,tabl
%O 1,8
%A _Wouter Meeussen_, Sep 16 2010
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