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A131632 Triangle T(n,k) read by rows = number of partitions of n-set into k blocks with distinct sizes, k = 1..A003056(n).

%I A131632 #38 May 06 2020 12:19:39
%S A131632 1,1,1,3,1,4,1,15,1,21,60,1,63,105,1,92,448,1,255,2016,1,385,4980,
%T A131632 12600,1,1023,15675,27720,1,1585,61644,138600,1,4095,155155,643500,1,
%U A131632 6475,482573,4408404,1,16383,1733550,12687675,37837800,1,26332,4549808,60780720
%N A131632 Triangle T(n,k) read by rows = number of partitions of n-set into k blocks with distinct sizes, k = 1..A003056(n).
%C A131632 Row sums = A007837.
%C A131632 Sum k! * T(n,k) = A032011.
%C A131632 Sum k * T(n,k) = A131623. - _Geoffrey Critzer_, Aug 30 2012.
%C A131632 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 A131632 Alois P. Heinz, <a href="/A131632/b131632.txt">Rows n = 1..500, flattened</a>
%F A131632 E.g.f.: Product_{n>=1} (1+y*x^n/n!).
%F A131632 T(A000217(n),n) = A022915(n). - _Alois P. Heinz_, Jul 03 2018
%e A131632 Triangle T(n,k)begins:
%e A131632   1;
%e A131632   1;
%e A131632   1,     3;
%e A131632   1,     4;
%e A131632   1,    15;
%e A131632   1,    21,      60;
%e A131632   1,    63,     105;
%e A131632   1,    92,     448;
%e A131632   1,   255,    2016;
%e A131632   1,   385,    4980,    12600;
%e A131632   1,  1023,   15675,    27720;
%e A131632   1,  1585,   61644,   138600;
%e A131632   1,  4095,  155155,   643500;
%e A131632   1,  6475,  482573,  4408404;
%e A131632   1, 16383, 1733550, 12687675, 37837800;
%e A131632   ...
%p A131632 b:= proc(n, i, t, v) option remember; `if`(t=1, 1/(n+v)!,
%p A131632       add(b(n-j, j, t-1, v+1)/(j+v)!, j=i..n/t))
%p A131632     end:
%p A131632 T:= (n, k)->`if`(k*(k+1)/2>n, 0, n!*b(n-k*(k+1)/2, 0, k, 1)):
%p A131632 seq(seq(T(n, k), k=1..floor(sqrt(2+2*n)-1/2)), n=1..20);
%p A131632 # _Alois P. Heinz_, Jun 21 2013
%p A131632 # second Maple program:
%p A131632 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A131632       `if`(n=0, 1, b(n, i-1)+binomial(n, i)*
%p A131632        expand(x*b(n-i, min(n-i, i-1)))))
%p A131632     end:
%p A131632 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
%p A131632 seq(T(n), n=1..20);  # _Alois P. Heinz_, Sep 27 2019
%t A131632 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 A131632 Columns k=1-10 give: A000012, A272514, A272515, A272516, A272517, A272518, A272519, A272520, A272521, A272522.
%Y A131632 Cf. A000217, A003056, A007837, A022915, A032011, A131623, A226873, A226874, A327803.
%K A131632 nonn,tabf
%O A131632 1,4
%A A131632 _Vladeta Jovovic_, Sep 04 2007