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A208741 Triangular array read by rows. T(n,k) is the number of sets of exactly k distinct binary words with a total of n letters.

%I A208741 #57 Feb 22 2023 09:54:50
%S A208741 2,4,1,8,8,16,22,4,32,64,20,64,156,84,6,128,384,264,40,256,888,784,
%T A208741 189,4,512,2048,2152,704,50,1024,4592,5664,2384,272,1,2048,10240,
%U A208741 14368,7328,1232,32,4096,22496,35568,21382,4704,248
%N A208741 Triangular array read by rows.  T(n,k) is the number of sets of exactly k distinct binary words with a total of n letters.
%C A208741 Equivalently, T(n,k) is the number of integer partitions of n into distinct parts with two types of 1's, four types of 2's, ... , 2^i types of i's,...; where k is the number of summands (of any type).
%C A208741 Row sums = A102866.
%C A208741 Row lengths increase by 1 at n=A061168(offset).
%H A208741 Alois P. Heinz, <a href="/A208741/b208741.txt">Rows n = 0..300, flattened</a>
%H A208741 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 64
%F A208741 O.g.f.: Product_{i>=1} (1 + y*x^i)^(2^i).
%e A208741 T(3,2) = 8 because we have: {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}; 2 word languages with total length 3.
%e A208741 Triangle T(n,k) begins:
%e A208741    2;
%e A208741    4,     1;
%e A208741    8,     8;
%e A208741   16,    22,    4;
%e A208741   32,    64,   20;
%e A208741   64,   156,   84,   6;
%e A208741   ...
%p A208741 h:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
%p A208741       add(h(n-i*j, i-1)*binomial(2^i, j)*x^j, j=0..n/i))))
%p A208741     end:
%p A208741 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(h(n$2)):
%p A208741 seq(T(n), n=1..15);  # _Alois P. Heinz_, Sep 24 2017
%t A208741 nn=12; p=Product[(1+y x^i)^(2^i), {i,1,nn}]; f[list_] := Select[list, #>0&]; Map[f, Drop[CoefficientList[Series[p[x,y], {x,0,nn}], {x,y}], 1]]//Flatten
%Y A208741 Cf. A102866, A209406, A360634.
%K A208741 nonn,tabf
%O A208741 1,1
%A A208741 _Geoffrey Critzer_, Mar 08 2012