OFFSET
1,3
LINKS
Alois P. Heinz, Rows n = 1..75, flattened
Wikipedia, Multinomial coefficients
EXAMPLE
T(3,2) = 2 = |{3!/(2!*1!), 3!/(1!*1!*1!)}| = |{3, 6}|.
T(5,2) = 3 = |{30, 60, 120}|.
T(7,4) = 10 = |{35, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040}|.
T(8,3) = 10 = |{560, 1120, 1680, 2520, 3360, 5040, 6720, 10080, 20160, 40320}|.
T(9,2) = 5 = |{22680, 45360, 90720, 181440, 362880}|.
Triangle T(n,k) begins:
1;
1, 2;
1, 2, 3;
1, 3, 4, 5;
1, 3, 5, 6, 7;
1, 4, 7, 9, 10, 11;
1, 4, 8, 10, 12, 13, 14;
1, 5, 10, 14, 17, 18, 19, 20;
1, 5, 12, 16, 21, 23, 25, 26, 27;
1, 6, 14, 20, 27, 29, 32, 34, 35, 36;
MAPLE
b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},
{b(n, i-1)[], seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)}))
end:
T:= (n, k)-> nops(b(n, k)):
seq(seq(T(n, k), k=1..n), n=1..14);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Join[b[n, i - 1], Table[ b[n - i*j, i - 1] *i!^j, {j, 1, n/i}] // Flatten]] // Union]; T[n_, k_] := Length[b[n, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 14 2012
STATUS
approved