A117195 Triangle read by rows: T(n,k) = number of partitions into distinct parts having rank k, 0<=k

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 2, 1, 1, 1, 0, 1, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1, 0, 1, 1, 3, 2, 3, 2, 2, 1, 1, 1, 0, 1, 0, 1, 2, 2, 4, 2, 3, 2, 2, 1, 1, 1, 0, 1

Offset: 1

Author

Reinhard Zumkeller, Mar 03 2006

Keywords

• nonn
• tabl

Comments

T(n,0) = A010054(n), T(n,1) = 1-A010054(n) for n>1;

A000009(n) = Sum(T(n,k): 0<=k

A117192(n) = Sum(T(n,k)*(1 - k mod 2): 0<=k

A117193(n) = Sum(T(n,k)*(k mod 2): 0<=k

A117194(n) = Sum(T(n,k)*(1 - k mod 2): 0

Examples

			Triangle starts:
[ 1]   1,
[ 2]   0, 1,
[ 3]   1, 0, 1,
[ 4]   0, 1, 0, 1,
[ 5]   0, 1, 1, 0, 1,
[ 6]   1, 0, 1, 1, 0, 1,
[ 7]   0, 1, 1, 1, 1, 0, 1,
[ 8]   0, 1, 1, 1, 1, 1, 0, 1,
[ 9]   0, 1, 1, 2, 1, 1, 1, 0, 1,
[10]   1, 0, 2, 1, 2, 1, 1, 1, 0, 1,
[11]   0, 1, 1, 2, 2, 2, 1, 1, 1, 0, 1,
[12]   0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1,
[13]   0, 1, 1, 3, 2, 3, 2, 2, 1, 1, 1, 0, 1,
[14]   0, 1, 2, 2, 4, 2, 3, 2, 2, 1, 1, 1, 0, 1, ...
T(12,0) = #{} = 0,
T(12,1) = #{5+4+2+1} = 1,
T(12,2) = #{6+3+2+1, 5+4+3} = 2,
T(12,3) = #{6+5+1, 6+4+2} = 2,
T(12,4) = #{7+4+1, 7+3+2} = 2,
T(12,5) = #{8+3+1, 7+5} = 2,
T(12,6) = #{9+2+1, 8+4} = 2,
T(12,7) = #{9+3} = 1,
T(12,8) = #{10+2} = 1,
T(12,9) = #{11+1} = 1,
T(12,10) = #{} = 0,
T(12,11) = #{12} = 1.
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Maria Monks, Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts, Proceedings of The American Mathematical Society, vol.138, no.02, pp.481-494, 2009.

Crossrefs

Cf. A063995, A105806.

Programs

Maple

b:= proc(n, i, k) option remember;
      if n<0 or k<0 then []
    elif n=0 then [0$k, 1]
    elif i<1 then []
    else zip ((x, y)-> x+y, b(n, i-1, k), b(n-i, i-1, k-1), 0)
      fi
    end:
T:= proc(n) local j, r; r:= [];
      for j from 0 to n do
        r:= zip ((x, y)-> x+y, r, b(n-j, j-1, j-1), 0)
      od; r[]
    end:
seq (T(n), n=1..20);  # Alois P. Heinz, Aug 29 2011

Mathematica

b[n_, i_, k_] := b[n, i, k] = Which[n<0 || k<0, {}, n == 0, Append[Array[0&, k], 1], i<1, {}, True, Plus @@ PadRight[{b[n, i-1, k], b[n-i, i-1, k-1]}]]; T[n_] := Module[{j, r}, r = {}; For[j = 0, j <= n, j++, r = Plus @@ PadRight[{r, b[n-j, j-1, j-1]}]]; r]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

PARI

N=33;  L=1+2*ceil(sqrtint(N));
q='q+O(q^N);
gf=sum(n=1,L, q^(n*(n+1)/2) / prod(k=1,n,1-z*q^k) );
v=Vec(gf);
{ for (n=1,#v,  /* print triangle: */
    p = Pol(v[n], 'z) + 'c0;
    p = polrecip(p);
    rw = Vec(p);  rw[1] -= 'c0;
    print1("[", n, "]   " );
    print( rw );
); }
/* Joerg Arndt, Oct 07 2012 */

Formula

G.f.: sum(n>=1, q^(n*(n+1)/2) / prod(k=1..n, 1-z*q^k) ), see Monks reference. [Joerg Arndt, Oct 07 2012]

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