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

Author

Wouter Meeussen, Apr 20 2013

Keywords

• nonn
• tabf

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).

Examples

			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.
		

Crossrefs

Cf. A000720, A077028.

Programs

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]}]