OFFSET
1,3
COMMENTS
Differs from A036038 after a(37). To illustrate where the difference comes from, consider 4,1,1,1 and 3,2,2 are two different partitions of 7 having the same value of multinomial coefficient M(n)=n!/(m1!*m2!*...*mk!)=210.
There is no known formula for M(n) sequence, however the asymptotic behavior has been studied, see the paper by Andrews, Knopfmacher, and Zimmermann.
The number of terms per row (for each value of n starting with n=1) forms sequence A070289.
LINKS
Alois P. Heinz, Rows n = 1..29, flattened
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the Number of Distinct Multinomial Coefficients, arXiv:math/0509470 [math.CO], 2005.
Sergei Viznyuk, C-program for the sequence
EXAMPLE
Trianglebegins:
1;
1, 2;
1, 3, 6;
1, 4, 6, 12, 24;
1, 5, 10, 20, 30, 60, 120;
1, 6, 15, 20, 30, 60, 90, 120, 180, 360, 720;
1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040;
...
Thus for n=4 (fourth row) the distinct values of multinomial coefficients are:
4!/(4!) = 1
4!/(3!1!) = 4
4!/(2!2!) = 6
4!/(2!1!1!) = 12
4!/(1!1!1!1!) = 24
MAPLE
b:= proc(n, i) option remember; `if`(n=0 or i<2, {1},
{seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=0..n/i)})
end:
T:= n-> sort([map(x-> n!/x, b(n, n))[]])[]:
seq(T(n), n=1..10); # Alois P. Heinz, Aug 13 2012
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0 || i<2, {1}, Union[Flatten @ Table[(#*i!^j&) /@ b[n-i*j, i-1], {j, 0, n/i}]]]; T[n_] := Sort[Flatten[n!/#& /@ b[n, n]] ]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Sergei Viznyuk, Mar 18 2012
STATUS
approved