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.

2, 4, 3, 8, 8, 4, 16, 26, 12, 5, 32, 64, 44, 16, 6, 64, 164, 132, 62, 20, 7, 128, 384, 376, 200, 80, 24, 8, 256, 904, 1008, 623, 268, 98, 28, 9, 512, 2048, 2632, 1792, 870, 336, 116, 32, 10, 1024, 4624, 6624, 5040, 2632, 1117, 404, 134, 36, 11

Offset: 1

Author

Geoffrey Critzer, Mar 08 2012

Keywords

• nonn
• tabl

Comments

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).

Row sums = A034899.

Examples

			Triangle T(n,k) begins:
    2;
    4,    3;
    8,    8,    4;
   16,   26,   12,    5;
   32,   64,   44,   16,   6;
   64,  164,  132,   62,  20,   7;
  128,  384,  376,  200,  80,  24,   8;
  256,  904, 1008,  623, 268,  98,  28,  9;
  512, 2048, 2632, 1792, 870, 336, 116, 32, 10;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Index entries for triangles generated by the Multiset Transformation

Crossrefs

Cf. A034899, A055375, A208741, A290222, A292506.

T(2n,n) gives A359962.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(2^i+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017

Mathematica

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

Formula

O.g.f.: Product_{i>=1} 1/(1-y*x^i)^(2^i).