A360626 - OEIS

%I #18 Nov 17 2023 11:35:14

%S 1,3,21,131,830,5066,30456,179256,1038593,5928071,33402561,186021335,

%T 1025162709,5596047683,30282832593,162573152651,866385400935,

%U 4585861723905,24120596727003,126124094912499,655868112470175,3393060517486981,17468543071082489

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

%H Alois P. Heinz, <a href="/A360626/b360626.txt">Table of n, a(n) for n = 0..380</a>

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

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

%e a(0) = 1: {}.

%e a(1) = 3: {ab}, {ba}, {a,b}.

%e 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}.

%p g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(

%p       g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))

%p     end:

%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,

%p      `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))

%p     end:

%p a:= n-> coeff(b(2*n$2), x, n):

%p seq(a(n), n=0..31);

%t 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}]]]];

%t 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}]]]];

%t a[n_] := Coefficient[b[2n, 2n], x, n];

%t Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, Nov 17 2023, after _Alois P. Heinz_ *)

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

%K nonn

%O 0,2

%A _Alois P. Heinz_, Feb 14 2023