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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

%I

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

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

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

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

%e 0 1 0

%e 1 0 1

%e 0 2 2

%e 0 2 0 1 3

%e 0 0 3 0 4

%e 0 3 0 2 5

%e 1 0 4 0 6

%e 0 4 0 3 7

%e 2 0 5 0 8

%e 0 5 0 4 9

%e 0 3 0 6 0 1 10

%e 0 0 6 0 5 0 11

%e 0 4 0 7 0 2 12

%e 0 0 7 0 6 0 13

%e 0 5 0 8 0 3 14

%e 1 0 8 0 7 0 15

%e ...

%e The table then starts as:

%e 0 0,1

%e 1 1,0

%e 2 0,2

%e 3 0,2,0,1

%e 4 0,0,5,0

%e 5 0,5,0,2

%e 6 1,0,10,0

%e 7 0,10,0,5

%e 8 2,0,20,0

%e 9 0,20,0,10

%e 10 0,5,0,36,0,1

%e ...

%e 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 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 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 Cf. A000720, A077028.

%K nonn,tabf

%O 0,6

%A _Wouter Meeussen_, Apr 20 2013

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Last modified June 24 13:44 EDT 2021. Contains 345417 sequences. (Running on oeis4.)