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

%I A224937 #15 Apr 15 2015 05:27:08
%S A224937 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,
%T A224937 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,
%U A224937 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
%N A224937 Number of partitions of n having T(n,k) odd parts in excess on even places over odd places.
%C A224937 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.
%C A224937 P. D. Hanna remarks that "zig-zag" diagonals/antidiagonals produce A077028 (Rascal triangle).
%e A224937 In the table below, replace each integer i with A000720(i) to get the  current sequence:
%e A224937 -3     -2      -1       0       1       2 (= k)(n= )
%e A224937                 0       1                         0
%e A224937                 1       0                         1
%e A224937                 0       2                         2
%e A224937         0       2       0       1                 3
%e A224937         0       0       3       0                 4
%e A224937         0       3       0       2                 5
%e A224937         1       0       4       0                 6
%e A224937         0       4       0       3                 7
%e A224937         2       0       5       0                 8
%e A224937         0       5       0       4                 9
%e A224937 0       3       0       6       0       1         10
%e A224937 0       0       6       0       5       0         11
%e A224937 0       4       0       7       0       2         12
%e A224937 0       0       7       0       6       0         13
%e A224937 0       5       0       8       0       3         14
%e A224937 1       0       8       0       7       0         15
%e A224937 ...
%e A224937 The table then starts as:
%e A224937 0  0,1
%e A224937 1  1,0
%e A224937 2  0,2
%e A224937 3  0,2,0,1
%e A224937 4  0,0,5,0
%e A224937 5  0,5,0,2
%e A224937 6  1,0,10,0
%e A224937 7  0,10,0,5
%e A224937 8  2,0,20,0
%e A224937 9  0,20,0,10
%e A224937 10 0,5,0,36,0,1
%e A224937   ...
%e A224937 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.
%t A224937 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 *)
%t A224937 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]}]
%Y A224937 Cf. A000720, A077028.
%K A224937 nonn,tabf
%O A224937 0,6
%A A224937 _Wouter Meeussen_, Apr 20 2013