OFFSET
1,40
COMMENTS
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)
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 *)
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 );
); }
/* Joerg Arndt, Oct 07 2012 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Mar 03 2006
STATUS
approved