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A292222 Triangle corresponding to the partition array of the M_1 multinomials (A036038). 1
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A212360

Adjacent sequences:  A292219 A292220 A292221 * A292223 A292224 A292225

KEYWORD

nonn,tabl,easy

AUTHOR

Wolfdieter Lang, Sep 29 2017

STATUS

approved

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Last modified March 24 11:34 EDT 2019. Contains 321448 sequences. (Running on oeis4.)