A292222 Triangle corresponding to the partition array of the M_1 multinomials (A036038).

1, 1, 2, 1, 3, 6, 1, 10, 12, 24, 1, 15, 50, 60, 120, 1, 41, 180, 300, 360, 720, 1, 63, 497, 1260, 2100, 2520, 5040, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320, 1, 255, 5154, 20916, 58464, 90720, 151200, 181440, 362880, 1, 637, 13680, 95640, 322560, 584640, 907200, 1512000, 1814400, 3628800

Offset: 1

Author

Wolfdieter Lang, Sep 29 2017

Keywords

• nonn
• tabl
• easy

Comments

Abramowitz-Stegun (A-St) M_1 multinomials as partition array (partitions in A-St order) are given in A036038. See this for details.

This is the sub-triangle of A226874(n,k) for n >= k >= 1 (here k=m).

The M_1 multinomials for a partition written in exponent form P = [1^e[1], 2^e[2], ... n^e[n]] with nonnegative e[j], for j =1, ..., n, is M_1(P) = n!/Product_{j=1..n} j!^e[j]. See the A-St link.

Examples

			The triangle T(n, m) begins:
n\m  1   2     3     4      5      6      7       8       9      10 ...
1:   1
2:   1   2
3:   1   3     6
4:   1  10    12    24
5:   1  15    50    60    120
6:   1  41   180   300    360    720
7:   1  63   497  1260   2100   2520   5040
8:   1 162  1484  6496  10080  16800  20160   40320
9:   1 255  5154 20916  58464  90720 151200  181440  362880
10:  1 637 13680 95640 322560 584640 907200 1512000 1814400 3628800
...
T(5, 3) =50 because the partitions are [1^2, 3^1] and [1^1, 2^2] with M_1 numbers 20 = A036038(5, 4) and 30 = A036038(5, 5), respectively, adding to 50.
		

Links

• Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.

Crossrefs

Cf. A036038, A130534 (M_2 triangle = |Stirling1|), A008277 (M_3 triangle = Stirling2), A226874 (M_1 triangle including empty partition).

Programs

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
t[n_, k_] := If[n*k == 0, If[n == k, 1, 0], n!*b[n, 1, k]];
Table[Table[t[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Sep 29 2017, after Alois P. Heinz *)

Formula

T(n, m) = sum over the A036038 entries in row n with parts number m, for m >= n >= 1.