OFFSET
1,5
LINKS
Alois P. Heinz, Antidiagonals n = 1..18, flattened
N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.
FORMULA
Row n is a polynomial in m: see A213074 for the coefficients.
EXAMPLE
The array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
1, 5, 14, 30, 55, 91, 140, 204, 285, 385, ...
1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, ...
1, 11, 53, 173, 448, 994, 1974, 3606, 6171, 10021, ...
1, 15, 89, 343, 1022, 2562, 5670, 11418, 21351, 37609, ...
1, 22, 167, 778, 2710, 7764, 19314, 43164, 88671, 170170, ...
...
MAPLE
with(combinat):
g:= proc(n, m) option remember;
`if`(m>1, map(x-> map(y-> sort([x[], y[]]), g(n, 1))[],
g(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x),
{partition(n)[]}), {[]}))
end:
T:= (n, m)-> nops(g(n, m)):
seq(seq(T(d-m, m), m=0..d-1), d=1..12); # Alois P. Heinz, Jul 11 2012
MATHEMATICA
T[n_, m_] := Module[{ip, lg, i}, ip = IntegerPartitions[n]; lg = Length[ ip]; i[0]=1; Table[Join[Sequence @@ Table[ip[[i[k]]], {k, 1, m}]] // Sort, Evaluate[Sequence @@ Table[{i[k], i[k-1], lg}, {k, 1, m}]]] // Flatten[#, m-1]& // Union // Length]; T[_, 0] = 1;
Table[T[n-m, m], {n, 1, 12}, {m, 0, n - 1}] // Flatten (* Jean-François Alcover, May 25 2016 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jun 05 2012
EXTENSIONS
More terms and cross-references from Alois P. Heinz, Jul 11 2012
STATUS
approved