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A240021
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Number T(n,k) of partitions of n into distinct parts, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, read by rows.
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21
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1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 0, 2, 2, 2, 4, 1, 0, 1, 2, 1, 1, 4, 2, 4, 5, 1, 1, 1, 1, 2, 1, 2, 6, 3, 1, 6, 6, 1, 2, 2, 1, 3, 1, 5, 9, 3, 2, 9, 7, 2, 4, 3, 2, 3, 2, 8, 12, 4, 0, 1, 4, 12, 8, 3, 7, 4, 3, 4, 3, 14, 16, 4, 1, 1, 7, 16, 9, 6, 11, 5, 1, 4, 4, 6, 20, 20, 5, 2, 2
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OFFSET
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0,10
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COMMENTS
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T(n,k) is defined for all n >= 0, k in A001057. Row n contains all terms from the leftmost to the rightmost nonzero term. All other terms (not in the triangle) are equal to 0. First nonzero term of column k>=0 is at n = k^2, first nonzero term of column k<=0 is at n = k*(k+1).
T(n,k) = T(n+k,-k).
T(n^2,n) = 1.
T(n^2,n-1) = 0.
Sum_{k=-n..n} T(n,k) = A000009 (row sums).
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LINKS
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FORMULA
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G.f.: prod(n>=1, 1 + e(n)*q^n ) = 1 + sum(n>=1, e(n)*q^n * prod(k=1..n-1, 1+e(k)*q^k) ) where e(n) = u if n odd, otherwise 1/u; see Pari program. [Joerg Arndt, Apr 01 2014]
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EXAMPLE
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T(12,-3) = 1: [6,4,2].
T(12,-2) = 2: [10,2], [8,4].
T(12,-1) = 1: [12].
T(12,0) = 2: [6,3,2,1], [5,4,2,1].
T(12,1) = 6: [9,2,1], [8,3,1], [7,4,1], [7,3,2], [6,5,1], [5,4,3].
T(12,2) = 3: [11,1], [9,3], [7,5].
T(13,-1) = 6: [10,2,1], [8,4,1], [8,3,2], [7,4,2], [6,5,2], [6,4,3].
T(14,-2) = 3: [12,2], [10,4], [8,6].
Triangle T(n,k) begins:
: n\k : -3 -2 -1 0 1 2 3 ...
+-----+--------------------------
: 0 : 1
: 1 : 1
: 2 : 1
: 3 : 1, 1
: 4 : 1, 0, 0, 1
: 5 : 2, 1
: 6 : 1, 1, 0, 1, 1
: 7 : 1, 3, 1
: 8 : 1, 1, 0, 2, 2
: 9 : 2, 4, 1, 0, 1
: 10 : 2, 1, 1, 4, 2
: 11 : 4, 5, 1, 1, 1
: 12 : 1, 2, 1, 2, 6, 3
: 13 : 1, 6, 6, 1, 2, 2
: 14 : 1, 3, 1, 5, 9, 3
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MAPLE
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b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
expand(b(n, i-1)+`if`(i>n, 0, b(n-i, i-1)*x^(2*irem(i, 2)-1)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2)):
seq(T(n), n=0..20);
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n>i*(i+1)/2, 0, If[n == 0, 1, Expand[b[n, i-1] + If[i>n, 0, b[n-i, i-1]*x^(2*Mod[i, 2]-1)]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n]]; Table[ T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)
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PROG
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(PARI)
N=20; q='q+O('q^N);
e(n) = if(n%2!=0, u, 1/u);
gf = prod(n=1, N, 1 + e(n)*q^n );
V = Vec( gf );
{ for (j=1, #V, \\ print triangle, including leading zeros
for (i=0, N-j, print1(" ")); \\ padding
for (i=-j+1, j-1, print1(polcoeff(V[j], i, u), ", "));
print();
); }
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CROSSREFS
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Columns k=0-10 give: A239241, A239871(n+1), A240138, A240139, A240140, A240141, A240142, A240143, A240144, A240145, A240146.
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KEYWORD
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AUTHOR
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STATUS
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approved
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