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

%I A292222 #19 Dec 07 2018 19:01:47
%S A292222 1,1,2,1,3,6,1,10,12,24,1,15,50,60,120,1,41,180,300,360,720,1,63,497,
%T A292222 1260,2100,2520,5040,1,162,1484,6496,10080,16800,20160,40320,1,255,
%U A292222 5154,20916,58464,90720,151200,181440,362880,1,637,13680,95640,322560,584640,907200,1512000,1814400,3628800
%N A292222 Triangle corresponding to the partition array of the M_1 multinomials (A036038).
%C A292222 Abramowitz-Stegun (A-St) M_1 multinomials as partition array (partitions in A-St order) are given in A036038. See this for details.
%C A292222 This is the sub-triangle of A226874(n,k) for n >= k >= 1 (here k=m).
%C A292222 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.
%H A292222 Milton Abramowitz and Irene A. Stegun, editors, <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=831&amp;Submit=Go">Multinomials: M_1, M_2 and M_3</a>, Handbook of Mathematical Functions, December 1972, pp. 831-2.
%F A292222 T(n, m) = sum over the A036038 entries in row n with parts number m, for m >= n >= 1.
%e A292222 The triangle T(n, m) begins:
%e A292222 n\m  1   2     3     4      5      6      7       8       9      10 ...
%e A292222 1:   1
%e A292222 2:   1   2
%e A292222 3:   1   3     6
%e A292222 4:   1  10    12    24
%e A292222 5:   1  15    50    60    120
%e A292222 6:   1  41   180   300    360    720
%e A292222 7:   1  63   497  1260   2100   2520   5040
%e A292222 8:   1 162  1484  6496  10080  16800  20160   40320
%e A292222 9:   1 255  5154 20916  58464  90720 151200  181440  362880
%e A292222 10:  1 637 13680 95640 322560 584640 907200 1512000 1814400 3628800
%e A292222 ...
%e A292222 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.
%t A292222 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 A292222 t[n_, k_] := If[n*k == 0, If[n == k, 1, 0], n!*b[n, 1, k]];
%t A292222 Table[Table[t[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Sep 29 2017, after _Alois P. Heinz_ *)
%Y A292222 Cf. A036038, A130534 (M_2 triangle = |Stirling1|), A008277 (M_3 triangle = Stirling2), A226874 (M_1 triangle including empty partition).
%K A292222 nonn,tabl,easy
%O A292222 1,3
%A A292222 _Wolfdieter Lang_, Sep 29 2017