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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
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
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
KEYWORD
nonn,tabl,easy
AUTHOR
Wolfdieter Lang, Sep 29 2017
STATUS
approved