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A242626
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Number T(n,k) of compositions of n, where k is the difference between the number of odd parts and the number of even parts, both counted without multiplicity; triangle T(n,k), n>=0, read by rows.
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13
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1, 1, 1, 0, 1, 2, 2, 2, 3, 1, 2, 11, 2, 3, 2, 2, 14, 8, 6, 6, 33, 14, 11, 5, 15, 43, 45, 20, 44, 82, 99, 25, 6, 14, 74, 141, 230, 41, 12, 202, 260, 451, 85, 26, 6, 22, 351, 514, 953, 148, 54, 24, 766, 1049, 1798, 355, 104, 18, 104, 1301, 2321, 3503, 751, 194
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OFFSET
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0,6
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COMMENTS
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T(n^2,n) = T(n^2+n,-n) = n! = A000142(n) for n>=0.
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LINKS
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EXAMPLE
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T(8,-1) = 15: [2,2,2,2], [1,1,2,4], [1,1,4,2], [1,2,1,4], [1,2,4,1], [1,4,1,2], [1,4,2,1], [2,1,1,4], [2,1,4,1], [2,4,1,1], [4,1,1,2], [4,1,2,1], [4,2,1,1], [4,4], [8].
Triangle T(n,k) begins:
: n\k : -3 -2 -1 0 1 2 3 ...
+-----+------------------------------------
: 0 : 1;
: 1 : 1;
: 2 : 1, 0, 1;
: 3 : 2, 2;
: 4 : 2, 3, 1, 2;
: 5 : 11, 2, 3;
: 6 : 2, 2, 14, 8, 6;
: 7 : 6, 33, 14, 11;
: 8 : 5, 15, 43, 45, 20;
: 9 : 44, 82, 99, 25, 6;
: 10 : 14, 74, 141, 230, 41, 12;
: 11 : 202, 260, 451, 85, 26;
: 12 : 6, 22, 351, 514, 953, 148, 54;
: 13 : 24, 766, 1049, 1798, 355, 104;
: 14 : 18, 104, 1301, 2321, 3503, 751, 194;
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MAPLE
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b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
expand(add(`if`(j=0, 1, x^(2*irem(i, 2)-1))*
b(n-i*j, i-1, p+j)/j!, j=0..n/i))))
end:
T:= n->(p->seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..20);
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MATHEMATICA
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b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, Expand[Sum[If[j==0, 1, x^(2*Mod[i, 2]-1)]*b[n-i*j, i-1, p+j]/j!, {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 17 2017, translated from Maple *)
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CROSSREFS
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Columns k=(-5)-5 give: A242836, A242837, A242838, A242839, A242840, A242821, A242841, A242842, A242843, A242844, A242845.
Cf. A242498 (compositions with multiplicity), A242618 (partitions without multiplicity).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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