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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 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 January 27 12:24 EST 2023. Contains 359840 sequences. (Running on oeis4.)