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A290222 Multiset transform of A011782, powers of 2: 1, 2, 4, 8, 16, ...

%I A290222 #28 Dec 07 2019 08:15:02
%S A290222 1,0,1,0,2,1,0,4,2,1,0,8,7,2,1,0,16,16,7,2,1,0,32,42,20,7,2,1,0,64,96,
%T A290222 54,20,7,2,1,0,128,228,140,59,20,7,2,1,0,256,512,360,156,59,20,7,2,1,
%U A290222 0,512,1160,888,422,162,59,20,7,2,1,0,1024,2560,2168,1088,442,162,59,20,7,2,1
%N A290222 Multiset transform of A011782, powers of 2: 1, 2, 4, 8, 16, ...
%C A290222 T(n,k) is the number of multisets of exactly k binary words with a total of n letters and each word beginning with 1. T(4,2) = 7: {1,100}, {1,101}, {1,110}, {1,111}, {10,10}, {10,11}, {11,11}. - _Alois P. Heinz_, Sep 18 2017
%H A290222 Alois P. Heinz, <a href="/A290222/b290222.txt">Rows n = 0..140, flattened</a>
%H A290222 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F A290222 G.f.: 1 / Product_{j>=1} (1-y*x^j)^(2^(j-1)). - _Alois P. Heinz_, Sep 18 2017
%e A290222 The triangle starts:
%e A290222 1;
%e A290222 0    1;
%e A290222 0    2    1;
%e A290222 0    4    2    1;
%e A290222 0    8    7    2    1;
%e A290222 0   16   16    7    2   1;
%e A290222 0   32   42   20    7   2   1;
%e A290222 0   64   96   54   20   7   2  1;
%e A290222 0  128  228  140   59  20   7  2  1;
%e A290222 0  256  512  360  156  59  20  7  2  1;
%e A290222 0  512 1160  888  422 162  59 20  7  2  1;
%e A290222 0 1024 2560 2168 1088 442 162 59 20  7  2  1;
%e A290222 (...)
%p A290222 b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
%p A290222       `if`(min(i, p)<1, 0, add(binomial(2^(i-1)+j-1, j)*
%p A290222          b(n-i*j, i-1, p-j), j=0..min(n/i, p)))))
%p A290222     end:
%p A290222 T:= (n, k)-> b(n$2, k):
%p A290222 seq(seq(T(n, k), k=0..n), n=0..14);  # _Alois P. Heinz_, Sep 12 2017
%t A290222 b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[Binomial[2^(i - 1) + j - 1, j] b[n - i j, i - 1, p - j], {j, 0, Min[n/i, p]}]]]];
%t A290222 T[n_, k_] := b[n, n, k];
%t A290222 Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 07 2019, after _Alois P. Heinz_ *)
%Y A290222 Cf. A034691 (row sums), A000007 (column k=0), A011782 (column k=1), A178945(n-1) (column k=2).
%Y A290222 The reverse of the n-th row converges to A034899.
%Y A290222 Cf. A000079, A209406, A292506.
%K A290222 nonn,tabl
%O A290222 0,5
%A A290222 _M. F. Hasler_, Jul 24 2017