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A208437 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} that have exactly k distinct block sizes.

%I A208437 #31 May 01 2017 14:30:39
%S A208437 1,2,2,3,5,10,2,50,27,116,60,2,560,315,142,1730,2268,282,6123,14742,
%T A208437 1073,30122,72180,12600,2,116908,464640,97020,32034,507277,2676366,
%U A208437 997920,2,2492737,16400098,8751600,136853,15328119,94209206,81225144,1527528,56182092,673282610,614128515,37837800
%N A208437 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} that have exactly k distinct block sizes.
%C A208437 Column 1 = A038041.
%C A208437 Column 2 = A088142.
%C A208437 Column 3 = A133118.
%C A208437 Row sums = A000110 (Bell numbers).
%C A208437 Row n has floor([sqrt(1+8n)-1]/2) terms (number of terms increases by one at each triangular number). - _Franklin T. Adams-Watters_, Feb 26 2012
%H A208437 Alois P. Heinz, <a href="/A208437/b208437.txt">Rows n = 1..220, flattened</a>
%H A208437 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 180.
%F A208437 E.g.f.: Product_{i>=1} 1 + y *(exp(x^i/i!)-1).
%F A208437 T(n*(n+1)/2,n) = A022915(n). - _Alois P. Heinz_, Apr 08 2016
%e A208437 :    1;
%e A208437 :    2;
%e A208437 :    2,      3;
%e A208437 :    5,     10;
%e A208437 :    2,     50;
%e A208437 :   27,    116,     60;
%e A208437 :    2,    560,    315;
%e A208437 :  142,   1730,   2268;
%e A208437 :  282,   6123,  14742;
%e A208437 : 1073,  30122,  72180,   12600;
%p A208437 with(combinat):
%p A208437 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A208437       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p A208437       b(n-i*j, i-1)*`if`(j=0, 1, x), j=0..n/i))))
%p A208437     end:
%p A208437 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
%p A208437 seq(T(n), n=1..16);  # _Alois P. Heinz_, Aug 21 2014
%t A208437 nn = 15; p = Product[1 + y (Exp[x^i/i!] - 1), {i, 1, nn}];f[list_] := Select[list, # > 0 &];
%t A208437 Map[f, Drop[ Range[0, nn]! CoefficientList[Series[p, {x, 0, nn}], {x, y}], 1]] // Flatten
%Y A208437 Cf. A022915, A116608, A218868.
%K A208437 nonn,tabf
%O A208437 1,2
%A A208437 _Geoffrey Critzer_, Feb 26 2012