A309992
Triangle T(n,k) whose n-th row lists in increasing order the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts; n >= 0, 1 <= k <= A000009(n), read by rows.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 4, 1, 5, 10, 1, 6, 15, 60, 1, 7, 21, 35, 105, 1, 8, 28, 56, 168, 280, 1, 9, 36, 84, 126, 252, 504, 1260, 1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600, 1, 11, 55, 165, 330, 462, 495, 1320, 2310, 4620, 6930, 27720
Offset: 0
For n = 5 there are 3 partitions of 5 into distinct parts: [5], [4,1], [3,2]. So row 5 contains M(5;5) = 1, M(5;4,1) = 5 and M(5;3,2) = 10.
Triangle T(n,k) begins:
1;
1;
1;
1, 3;
1, 4;
1, 5, 10;
1, 6, 15, 60;
1, 7, 21, 35, 105;
1, 8, 28, 56, 168, 280;
1, 9, 36, 84, 126, 252, 504, 1260;
1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600;
1, 11, 55, 165, 330, 462, 495, 1320, 2310, 4620, 6930, 27720;
...
Rightmost terms of rows give
A290517.
-
g:= proc(n, i) option remember; `if`(i*(i+1)/2binomial(n, i)*x, g(n-i, min(n-i, i-1)))[], g(n, i-1)[]]))
end:
T:= n-> sort(g(n$2))[]:
seq(T(n), n=0..14);
-
g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, {}, If[n == 0, {1}, Join[ Binomial[n, i] # & /@ g[n-i, Min[n-i, i-1]], g[n, i-1]]]];
T[n_] := Sort[g[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Jan 27 2021, after Alois P. Heinz *)
A325593
Numbers having at least three representations as multinomial coefficients M(n;lambda), where lambda is a partition of n.
Original entry on oeis.org
1, 6, 20, 30, 56, 60, 90, 105, 120, 210, 252, 360, 420, 462, 495, 504, 560, 630, 720, 756, 840, 990, 1260, 1320, 1365, 1540, 1680, 1716, 1980, 2520, 2970, 3003, 3360, 3960, 4290, 4620, 5040, 5460, 6006, 6435, 7140, 7560, 7920, 8190, 9240, 10080, 10296, 10626
Offset: 1
1 is in the sequence because M(0;0) = M(1;1) = M(2;2) = M(3;3) = ... = 1.
6 is in the sequence because M(6;5,1) = M(4;2,2) = M(3;1,1,1) = 6.
56 is in the sequence because M(56;55,1) = M(8;5,3) = M(8;6,1,1) = 56.
A325901
Numbers having at least two representations as multinomial coefficients M(n;lambda), where lambda is a partition of n into distinct parts.
Original entry on oeis.org
1, 10, 15, 21, 28, 35, 36, 45, 55, 56, 60, 66, 78, 84, 91, 105, 120, 126, 136, 153, 165, 168, 171, 190, 210, 220, 231, 252, 253, 276, 280, 286, 300, 325, 330, 351, 360, 364, 378, 406, 435, 455, 462, 465, 495, 496, 504, 528, 560, 561, 595, 630, 660, 666, 680
Offset: 1
1 is in the sequence because M(0;0) = M(1;1) = M(2;2) = M(3;3) = ... = 1.
10 is in the sequence because M(10;9,1) = M(5;3,2) = 10.
55 is in the sequence because M(55;54,1) = M(11;9,2) = 55.
105 is in the sequence because M(7;4,2,1) = M(15;13,2) = M(105;104,1) = 105.
Showing 1-3 of 3 results.
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