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A209406 Triangular array read by rows: T(n,k) is the number of multisets of exactly k nonempty binary words with a total of n letters.

%I A209406 #31 Feb 22 2023 10:05:49
%S A209406 2,4,3,8,8,4,16,26,12,5,32,64,44,16,6,64,164,132,62,20,7,128,384,376,
%T A209406 200,80,24,8,256,904,1008,623,268,98,28,9,512,2048,2632,1792,870,336,
%U A209406 116,32,10,1024,4624,6624,5040,2632,1117,404,134,36,11
%N A209406 Triangular array read by rows: T(n,k) is the number of multisets of exactly k nonempty binary words with a total of n letters.
%C A209406 Equivalently, T(n,k) is the number of partitions of the integer n with two types of 1's, four types of 2's, ..., 2^i types of i's...; having exactly k summands (of any type).
%C A209406 Row sums = A034899.
%H A209406 Alois P. Heinz, <a href="/A209406/b209406.txt">Rows n = 1..141, flattened</a>
%H A209406 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F A209406 O.g.f.: Product_{i>=1} 1/(1-y*x^i)^(2^i).
%e A209406 Triangle T(n,k) begins:
%e A209406     2;
%e A209406     4,    3;
%e A209406     8,    8,    4;
%e A209406    16,   26,   12,    5;
%e A209406    32,   64,   44,   16,   6;
%e A209406    64,  164,  132,   62,  20,   7;
%e A209406   128,  384,  376,  200,  80,  24,   8;
%e A209406   256,  904, 1008,  623, 268,  98,  28,  9;
%e A209406   512, 2048, 2632, 1792, 870, 336, 116, 32, 10;
%e A209406   ...
%p A209406 b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
%p A209406       `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
%p A209406        binomial(2^i+j-1, j), j=0..min(n/i, p)))))
%p A209406     end:
%p A209406 T:= (n, k)-> b(n$2, k):
%p A209406 seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Apr 13 2017
%t A209406 nn = 10; p[x_, y_] := Product[1/(1 - y x^i)^(2^i), {i, 1, nn}]; f[list_] := Select[lst, # > 0 &]; Map[f, Drop[CoefficientList[Series[p[x, y], {x, 0, nn}], {x, y}], 1]] // Flatten
%Y A209406 Cf. A034899, A055375, A208741, A290222, A292506.
%Y A209406 T(2n,n) gives A359962.
%K A209406 nonn,tabl
%O A209406 1,1
%A A209406 _Geoffrey Critzer_, Mar 08 2012