A224937 Number of partitions of n having T(n,k) odd parts in excess on even places over odd places.

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

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