A117195 - OEIS

A117195

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

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,40

COMMENTS

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

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

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

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

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

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.

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]

EXAMPLE

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.

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)

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 *)

PROG

(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 );