A240021 - OEIS

%I #42 Feb 11 2015 04:09:56

%S 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,

%T 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,

%U 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

%N 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.

%C 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).

%C T(n,k) = T(n+k,-k).

%C T(2n*(2n+1),2n) = A000041(n).

%C T(4n^2+14n+11,2n+2) = A000070(n).

%C T(n^2,n) = 1.

%C T(n^2,n-1) = 0.

%C T(n^2,n-2) = A209815(n+1).

%C T(n^2+1,n-1) = A000065(n).

%C T(n,0) = A239241(n).

%C Sum_{k<=-1} T(n,k) = A239239(n).

%C Sum_{k<=0} T(n,k) = A239240(n).

%C Sum_{k>=1} T(n,k) = A239242(n).

%C Sum_{k>=0} T(n,k) = A239243(n).

%C Sum_{k=-1..1} T(n,k) = A239881(n).

%C T(n,-1) + T(n,1) = A239880(n).

%C Sum_{k=-n..n} T(n,k) = A000009 (row sums).

%H Alois P. Heinz, <a href="/A240021/b240021.txt">Rows n = 0..500, flattened</a>

%F 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]

%e T(12,-3) = 1: [6,4,2].

%e T(12,-2) = 2: [10,2], [8,4].

%e T(12,-1) = 1: [12].

%e T(12,0) = 2: [6,3,2,1], [5,4,2,1].

%e T(12,1) = 6: [9,2,1], [8,3,1], [7,4,1], [7,3,2], [6,5,1], [5,4,3].

%e T(12,2) = 3: [11,1], [9,3], [7,5].

%e T(13,-1) = 6: [10,2,1], [8,4,1], [8,3,2], [7,4,2], [6,5,2], [6,4,3].

%e T(14,-2) = 3: [12,2], [10,4], [8,6].

%e Triangle T(n,k) begins:

%e : n\k : -3 -2 -1  0  1  2  3  ...

%e +-----+--------------------------

%e :  0  :           1

%e :  1  :              1

%e :  2  :        1

%e :  3  :           1, 1

%e :  4  :        1, 0, 0, 1

%e :  5  :           2, 1

%e :  6  :     1, 1, 0, 1, 1

%e :  7  :        1, 3, 1

%e :  8  :     1, 1, 0, 2, 2

%e :  9  :        2, 4, 1, 0, 1

%e : 10  :     2, 1, 1, 4, 2

%e : 11  :        4, 5, 1, 1, 1

%e : 12  :  1, 2, 1, 2, 6, 3

%e : 13  :     1, 6, 6, 1, 2, 2

%e : 14  :  1, 3, 1, 5, 9, 3

%p b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,

%p       expand(b(n, i-1)+`if`(i>n, 0, b(n-i, i-1)*x^(2*irem(i, 2)-1)))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2)):

%p seq(T(n), n=0..20);

%t 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_ *)

%o (PARI)

%o N=20; q='q+O('q^N);

%o e(n) = if(n%2!=0, u, 1/u);

%o gf = prod(n=1,N, 1 + e(n)*q^n );

%o V = Vec( gf );

%o { for (j=1, #V,  \\ print triangle, including leading zeros

%o     for (i=0, N-j, print1("   "));  \\ padding

%o     for (i=-j+1, j-1, print1(polcoeff(V[j], i, u),", "));

%o     print();

%o ); }

%o /* _Joerg Arndt_, Apr 01 2014 */

%Y Columns k=0-10 give: A239241, A239871(n+1), A240138, A240139, A240140, A240141, A240142, A240143, A240144, A240145, A240146.

%Y Cf. A240009.

%K nonn,tabf,look

%O 0,10

%A _Alois P. Heinz_, Mar 31 2014