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A360693 Number T(n,k) of sets of n words of length n over binary alphabet where the first letter occurs k times; triangle T(n,k), n>=0, n-signum(n)<=k<=n*(n-1)+signum(n), read by rows.

%I A360693 #36 May 26 2023 03:49:06
%S A360693 1,1,1,2,2,2,3,10,15,15,10,3,4,37,108,228,336,394,336,228,108,37,4,5,
%T A360693 101,600,2150,5645,11680,19752,27820,32935,32935,27820,19752,11680,
%U A360693 5645,2150,600,101,5,6,226,2490,14745,61770,200529,535674,1211485,2368200
%N A360693 Number T(n,k) of sets of n words of length n over binary alphabet where the first letter occurs k times; triangle T(n,k), n>=0, n-signum(n)<=k<=n*(n-1)+signum(n), read by rows.
%C A360693 T(n,k) is defined for all n >= 0 and k >= 0.  The triangle contains only the positive elements.
%H A360693 Alois P. Heinz, <a href="/A360693/b360693.txt">Rows n = 0..33, flattened</a>
%F A360693 T(n,k) = T(n,n^2-k).
%e A360693 T(2,3) = 2: {aa,ab}, {aa,ba}.
%e A360693 T(3,3) = 10: {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}.
%e A360693 T(4,3) = 4: {abbb,babb,bbab,bbbb}, {abbb,babb,bbba,bbbb}, {abbb,bbab,bbba,bbbb}, {babb,bbab,bbba,bbbb}.
%e A360693 Triangle T(n,k) begins:
%e A360693   1;
%e A360693   1, 1;
%e A360693   .  2, 2,  2;
%e A360693   .  .  3, 10, 15,  15,  10,    3;
%e A360693   .  .  .   4, 37, 108, 228,  336,  394,   336,   228,   108,    37,     4;
%e A360693   .  .  .   .   5, 101, 600, 2150, 5645, 11680, 19752, 27820, 32935, 32935, ...;
%e A360693   ...
%p A360693 g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
%p A360693       g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i), k), k=0..j))))
%p A360693     end:
%p A360693 T:= n-> (p-> seq(coeff(p, x, i), i=n-signum(n)..n*(n-1)+signum(n)))(g(n$3)):
%p A360693 seq(T(n), n=0..6);
%t A360693 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 A360693 T[n_] := Table[Coefficient[#, x, i], {i, n - Sign[n], n(n - 1) + Sign[n]}]&[g[n, n, n]];
%t A360693 Table[T[n], {n, 0, 6}] // Flatten (* _Jean-François Alcover_, May 26 2023, after _Alois P. Heinz_ *)
%Y A360693 Row sums give A014070.
%Y A360693 Column sums give A360695.
%Y A360693 Main diagonal T(n,n) gives A154323(n-1) for n>=1.
%Y A360693 T(n,n-1) gives A000027(n) for n>=1.
%Y A360693 T(2n,2n^2) gives A360702.
%Y A360693 Cf. A000290, A057427, A220886 (similar triangle for multisets).
%K A360693 nonn,tabf
%O A360693 0,4
%A A360693 _Alois P. Heinz_, Feb 16 2023