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A224937 Number of partitions of n having T(n,k) odd parts in excess on even places over odd places. 0
0, 1, 1, 0, 0, 2, 0, 2, 0, 1, 0, 0, 5, 0, 0, 5, 0, 2, 1, 0, 10, 0, 0, 10, 0, 5, 2, 0, 20, 0, 0, 20, 0, 10, 0, 5, 0, 36, 0, 1, 0, 0, 36, 0, 20, 0, 0, 10, 0, 65, 0, 2, 0, 0, 65, 0, 36, 0, 0, 20, 0, 110, 0, 5, 1, 0, 110, 0, 65, 0, 0, 36, 0, 185, 0, 10, 2, 0, 185, 0, 110, 0, 0, 65, 0, 300, 0, 20 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Row lengths are 2*floor((3 + sqrt(1+8*n))/4), k runs from -floor((3 + sqrt(1+8*n))/4) up to floor((-1 + sqrt(1+8*n))/4); row sums are A000041.

P. D. Hanna remarks that "zig-zag" diagonals/antidiagonals produce A077028 (Rascal triangle).

LINKS

Table of n, a(n) for n=0..87.

EXAMPLE

In the table below, replace each integer i with A000720(i) to get the current sequence:

-3 -2 -1 0 1 2 (= k)(n= )

0 1 0

1 0 1

0 2 2

0 2 0 1 3

0 0 3 0 4

0 3 0 2 5

1 0 4 0 6

0 4 0 3 7

2 0 5 0 8

0 5 0 4 9

0 3 0 6 0 1 10

0 0 6 0 5 0 11

0 4 0 7 0 2 12

0 0 7 0 6 0 13

0 5 0 8 0 3 14

1 0 8 0 7 0 15

...

The table then starts as:

0 0,1

1 1,0

2 0,2

3 0,2,0,1

4 0,0,5,0

5 0,5,0,2

6 1,0,10,0

7 0,10,0,5

8 2,0,20,0

9 0,20,0,10

10 0,5,0,36,0,1

...

The partitions of n=5 then give (0,5,0,2) for k=(-2,-1,0,1); this corresponds to 5 partitions with -1 excess odd parts on even over odd positions, and 2 with 1 excess, namely (4,1') and (2,1',1,1') where odd parts on even positions are marked by a quote.

MATHEMATICA

Table[ CoefficientList[ x^Floor[(3+Sqrt[1+8*n])/4]* Tr[x^Tr[(-1)^Mod[Flatten[Position[#, _?OddQ]], 2]]&/@Partitions[n]], x], {n, 0, 12}]; (* or *)

a712[n_Integer]:= a712[n] =If[n<0, 0, (# . Reverse[#])& [PartitionsP[ Range[0, n] ]]]; Table[If[Mod[n+k, 2]==1, 0, a712[-1+Max[0, (2+n-k*(2*k+1))/2]]], {n, 0, 12}, {k, -Floor[(3+Sqrt[1+8*n])/4], Floor[(-1+Sqrt[1+8*n])/4]}]

CROSSREFS

Cf. A000720, A077028.

Sequence in context: A324735 A219495 A242041 * A035193 A004556 A263635

Adjacent sequences: A224934 A224935 A224936 * A224938 A224939 A224940

KEYWORD

nonn,tabf

AUTHOR

Wouter Meeussen, Apr 20 2013

STATUS

approved

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Last modified January 27 12:24 EST 2023. Contains 359840 sequences. (Running on oeis4.)