A360695 Total number of sets of k words of length k over binary alphabet with exactly n occurrences of the first letter in the set, summed over all k >= 0.

2, 3, 5, 16, 57, 230, 1071, 5429, 29810, 175718, 1101090, 7294593, 50829712, 370975443, 2826022446, 22403032310, 184339146428, 1570830751662, 13835026646912, 125719891784479, 1176838995406439, 11331919317891519, 112100167281082176, 1137938904082103310

Offset: 0

Author

Alois P. Heinz, Feb 16 2023

Keywords

• nonn

Comments

Also the number of inequivalent square {0,1} matrices with distinct rows and exactly n 1's modulo permutation of the rows. a(2) = 5: [11/00], [10/01], [100/010/000], [100/001/000], [010/001/000].

Examples

			a(0) = 2: {}, {b}.
a(1) = 3: {a}, {ab,bb}, {ba,bb}.
a(3) = 16: {aa,ab}, {aa,ba}, {aab,abb,bbb}, {aab,bab,bbb}, {aab,bba,bbb}, {aba,abb,bbb}, {aba,bab,bbb}, {aba,bba,bbb}, {abb,baa,bbb}, {abb,bab,bba}, {baa,bab,bbb}, {baa,bba,bbb}, {abbb,babb,bbab,bbbb}, {abbb,babb,bbba,bbbb}, {abbb,bbab,bbba,bbbb}, {babb,bbab,bbba,bbbb}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Column sums of A360693.

Programs

Maple

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), k=0..j))))
    end:
a:= n-> add(coeff(g(k$3), x, n), k=0..n+1):
seq(a(n), n=0..23);

Mathematica

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], {k, 0, j}]]]];
a[n_] := Sum[Coefficient[g[k, k, k], x, n], {k, 0, n + 1}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n+1} A360693(k,n).