%I #38 May 06 2020 12:19:39

%S 1,1,1,3,1,4,1,15,1,21,60,1,63,105,1,92,448,1,255,2016,1,385,4980,

%T 12600,1,1023,15675,27720,1,1585,61644,138600,1,4095,155155,643500,1,

%U 6475,482573,4408404,1,16383,1733550,12687675,37837800,1,26332,4549808,60780720

%N Triangle T(n,k) read by rows = number of partitions of n-set into k blocks with distinct sizes, k = 1..A003056(n).

%C Row sums = A007837.

%C Sum k! * T(n,k) = A032011.

%C Sum k * T(n,k) = A131623. - _Geoffrey Critzer_, Aug 30 2012.

%C T(n,k) is also the number of words w of length n over a k-ary alphabet {a1,a2,...,ak} with #(w,a1) > #(w,a2) > ... > #(w,ak) > 0, where #(w,x) counts the letters x in word w. T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa. - _Alois P. Heinz_, Jun 21 2013

%H Alois P. Heinz, <a href="/A131632/b131632.txt">Rows n = 1..500, flattened</a>

%F E.g.f.: Product_{n>=1} (1+y*x^n/n!).

%F T(A000217(n),n) = A022915(n). - _Alois P. Heinz_, Jul 03 2018

%e Triangle T(n,k)begins:

%e 1;

%e 1;

%e 1, 3;

%e 1, 4;

%e 1, 15;

%e 1, 21, 60;

%e 1, 63, 105;

%e 1, 92, 448;

%e 1, 255, 2016;

%e 1, 385, 4980, 12600;

%e 1, 1023, 15675, 27720;

%e 1, 1585, 61644, 138600;

%e 1, 4095, 155155, 643500;

%e 1, 6475, 482573, 4408404;

%e 1, 16383, 1733550, 12687675, 37837800;

%e ...

%p b:= proc(n, i, t, v) option remember; `if`(t=1, 1/(n+v)!,

%p add(b(n-j, j, t-1, v+1)/(j+v)!, j=i..n/t))

%p end:

%p T:= (n, k)->`if`(k*(k+1)/2>n, 0, n!*b(n-k*(k+1)/2, 0, k, 1)):

%p seq(seq(T(n, k), k=1..floor(sqrt(2+2*n)-1/2)), n=1..20);

%p # _Alois P. Heinz_, Jun 21 2013

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,

%p `if`(n=0, 1, b(n, i-1)+binomial(n, i)*

%p expand(x*b(n-i, min(n-i, i-1)))))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):

%p seq(T(n), n=1..20); # _Alois P. Heinz_, Sep 27 2019

%t nn=10;p=Product[1+y x^i/i!,{i,1,nn}];Range[0,nn]! CoefficientList[ Series[p,{x,0,nn}],{x,y}]//Grid (* _Geoffrey Critzer_, Aug 30 2012 *)

%Y Columns k=1-10 give: A000012, A272514, A272515, A272516, A272517, A272518, A272519, A272520, A272521, A272522.

%Y Cf. A000217, A003056, A007837, A022915, A032011, A131623, A226873, A226874, A327803.

%K nonn,tabf

%O 1,4

%A _Vladeta Jovovic_, Sep 04 2007