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%I #22 Apr 13 2022 13:01:25
%S 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,
%T 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,
%U 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
%N Triangle read by rows: T(n,k) = number of partitions into distinct parts having rank k, 0<=k<n.
%C T(n,0) = A010054(n), T(n,1) = 1-A010054(n) for n>1;
%C A000009(n) = Sum(T(n,k): 0<=k<n);
%C A117192(n) = Sum(T(n,k)*(1 - k mod 2): 0<=k<n);
%C A117193(n) = Sum(T(n,k)*(k mod 2): 0<=k<n);
%C A117194(n) = Sum(T(n,k)*(1 - k mod 2): 0<k<n);
%H Alois P. Heinz, <a href="/A117195/b117195.txt">Rows n = 1..141, flattened</a>
%H Maria Monks, <a href="https://doi.org/10.1090/S0002-9939-09-10076-X">Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts</a>, Proceedings of The American Mathematical Society, vol.138, no.02, pp.481-494, 2009.
%F 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]
%e Triangle starts:
%e [ 1] 1,
%e [ 2] 0, 1,
%e [ 3] 1, 0, 1,
%e [ 4] 0, 1, 0, 1,
%e [ 5] 0, 1, 1, 0, 1,
%e [ 6] 1, 0, 1, 1, 0, 1,
%e [ 7] 0, 1, 1, 1, 1, 0, 1,
%e [ 8] 0, 1, 1, 1, 1, 1, 0, 1,
%e [ 9] 0, 1, 1, 2, 1, 1, 1, 0, 1,
%e [10] 1, 0, 2, 1, 2, 1, 1, 1, 0, 1,
%e [11] 0, 1, 1, 2, 2, 2, 1, 1, 1, 0, 1,
%e [12] 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1,
%e [13] 0, 1, 1, 3, 2, 3, 2, 2, 1, 1, 1, 0, 1,
%e [14] 0, 1, 2, 2, 4, 2, 3, 2, 2, 1, 1, 1, 0, 1, ...
%e T(12,0) = #{} = 0,
%e T(12,1) = #{5+4+2+1} = 1,
%e T(12,2) = #{6+3+2+1, 5+4+3} = 2,
%e T(12,3) = #{6+5+1, 6+4+2} = 2,
%e T(12,4) = #{7+4+1, 7+3+2} = 2,
%e T(12,5) = #{8+3+1, 7+5} = 2,
%e T(12,6) = #{9+2+1, 8+4} = 2,
%e T(12,7) = #{9+3} = 1,
%e T(12,8) = #{10+2} = 1,
%e T(12,9) = #{11+1} = 1,
%e T(12,10) = #{} = 0,
%e T(12,11) = #{12} = 1.
%p b:= proc(n, i, k) option remember;
%p if n<0 or k<0 then []
%p elif n=0 then [0$k, 1]
%p elif i<1 then []
%p else zip ((x, y)-> x+y, b(n, i-1, k), b(n-i, i-1, k-1), 0)
%p fi
%p end:
%p T:= proc(n) local j, r; r:= [];
%p for j from 0 to n do
%p r:= zip ((x, y)-> x+y, r, b(n-j, j-1, j-1), 0)
%p od; r[]
%p end:
%p seq (T(n), n=1..20); # _Alois P. Heinz_, Aug 29 2011
%t 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_ *)
%o (PARI)
%o N=33; L=1+2*ceil(sqrtint(N));
%o q='q+O(q^N);
%o gf=sum(n=1,L, q^(n*(n+1)/2) / prod(k=1,n,1-z*q^k) );
%o v=Vec(gf);
%o { for (n=1,#v, /* print triangle: */
%o p = Pol(v[n], 'z) + 'c0;
%o p = polrecip(p);
%o rw = Vec(p); rw[1] -= 'c0;
%o print1("[", n, "] " );
%o print( rw );
%o ); }
%o /* _Joerg Arndt_, Oct 07 2012 */
%Y Cf. A063995, A105806.
%K nonn,tabl
%O 1,40
%A _Reinhard Zumkeller_, Mar 03 2006
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