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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.

%I A360695 #25 Dec 13 2023 14:43:50
%S A360695 2,3,5,16,57,230,1071,5429,29810,175718,1101090,7294593,50829712,
%T A360695 370975443,2826022446,22403032310,184339146428,1570830751662,
%U A360695 13835026646912,125719891784479,1176838995406439,11331919317891519,112100167281082176,1137938904082103310
%N 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.
%C A360695 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].
%H A360695 Alois P. Heinz, <a href="/A360695/b360695.txt">Table of n, a(n) for n = 0..200</a>
%F A360695 a(n) = Sum_{k=0..n+1} A360693(k,n).
%e A360695 a(0) = 2: {}, {b}.
%e A360695 a(1) = 3: {a}, {ab,bb}, {ba,bb}.
%e A360695 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}.
%p A360695 g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
%p A360695       g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i) , k), k=0..j))))
%p A360695     end:
%p A360695 a:= n-> add(coeff(g(k$3), x, n), k=0..n+1):
%p A360695 seq(a(n), n=0..23);
%t A360695 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}]]]];
%t A360695 a[n_] := Sum[Coefficient[g[k, k, k], x, n], {k, 0, n + 1}];
%t A360695 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Dec 13 2023, after _Alois P. Heinz_ *)
%Y A360695 Column sums of A360693.
%K A360695 nonn
%O A360695 0,1
%A A360695 _Alois P. Heinz_, Feb 16 2023