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

1,3

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.

Links

Table of n, a(n) for n=1..55.

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

Formula

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A289656 A248686 A059434 * A182928 A141476 A340114

Adjacent sequences: A292219 A292220 A292221 * A292223 A292224 A292225

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang, Sep 29 2017

Status

approved