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%I #25 Dec 07 2019 08:14:39
%S 1,1,1,2,2,1,5,5,4,1,15,15,14,7,1,52,52,51,36,11,1,203,203,202,171,81,
%T 16,1,877,877,876,813,512,162,22,1,4140,4140,4139,4012,3046,1345,295,
%U 29,1,21147,21147,21146,20891,17866,10096,3145,499,37,1,115975,115975,115974,115463,106133,72028,29503,6676,796,46,1
%N A triangular array of numbers related to factorization and number of parts in Murasaki diagrams.
%C When the Bell multisets are encoded as described in A130274, the seven case in the example can be coded as 19578, 15942, 30873, 26427, 35642, 29491 and 32938.
%H Robert Israel, <a href="/A133611/b133611.txt">Table of n, a(n) for n = 1..10011</a>
%F Equals A048993 * A000012. - _Gary W. Adamson_, Jan 29 2008
%F That is, T(i,j) = Sum_{k=j..i} A048993(i,k) for 0 <= j <= i. - _Robert Israel_, Nov 01 2018
%e The array begins:
%e 1
%e 1 1
%e 2 2 1
%e 5 5 4 1
%e 15 15 14 7 1
%e 52 52 51 36 11 1
%e ...
%e a(14) = 7 because only seven of the 52 Bell multisets can be generated by attaching a new stroke to the third element in the set of diaqrams with four strokes.
%p T:= proc(i,j) add(combinat:-stirling2(i,k),k=j..i) end proc:
%p seq(seq(T(i,j),j=0..i),i=0..15); # _Robert Israel_, Nov 01 2018
%p # second Maple program:
%p b:= proc(n, t) option remember; `if`(n>0, add(b(n-j, t+1)*
%p binomial(n-1, j-1), j=1..n), add(x^j, j=0..t))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Aug 30 2019
%t row[n_] := Table[StirlingS2[n, k], {k, 0, n}] // Reverse // Accumulate // Reverse;
%t Array[row, 11, 0] // Flatten (* _Jean-François Alcover_, Dec 07 2019 *)
%Y Cf. A000110 (row sums), A137650 (a similar triangle), A130274, A211561.
%Y Cf. A048993.
%K nonn,tabl
%O 1,4
%A _Alford Arnold_, Sep 18 2007
%E Definition not clear to me - _N. J. A. Sloane_, Sep 18 2007
%E More terms from _Robert Israel_, Nov 01 2018
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