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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 April 22 22:19 EDT 2021. Contains 343197 sequences. (Running on oeis4.)