A359962 -id:A359962 - cp's OEIS frontend

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

Geoffrey Critzer, Mar 08 2012

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.

			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;
  ...

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

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

T(2n,n) gives A359962.

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

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

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

A360626 Number of multisets of nonempty words over binary alphabet where each letter occurs n times.

Original entry on oeis.org

1, 3, 21, 131, 830, 5066, 30456, 179256, 1038593, 5928071, 33402561, 186021335, 1025162709, 5596047683, 30282832593, 162573152651, 866385400935, 4585861723905, 24120596727003, 126124094912499, 655868112470175, 3393060517486981, 17468543071082489

Offset: 0

Alois P. Heinz, Feb 14 2023

nonn

			a(0) = 1: {}.
a(1) = 3: {ab}, {ba}, {a,b}.
a(2) = 21: {aabb}, {abab}, {abba}, {baab}, {baba}, {bbaa}, {a,abb}, {a,bab}, {a,bba}, {aa,bb}, {aab,b}, {ab,ab}, {ab,ba}, {aba,b}, {b,baa}, {ba,ba}, {a,a,bb}, {a,ab,b}, {a,b,ba}, {aa,b,b}, {a,a,b,b}.

Alois P. Heinz, Table of n, a(n) for n = 0..380

Cf. A055375, A359962, A360638 (the same for sets).

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..31);

g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i] + k - 1, k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
a[n_] := Coefficient[b[2n, 2n], x, n];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)

a(n) = [x^(2n)*y^n] Product_{i>=1} Product_{j=0..i} 1/(1-x^i*y^j)^binomial(i,j).

a(n) = A055375(2n,n).

cp's OEIS Frontend

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.

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