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A320264 Number T(n,k) of proper multisets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the multiset; triangle T(n,k), n>=2, 1<=k<=n-1, read by rows.

%I A320264 #17 Feb 13 2021 14:04:58
%S A320264 1,1,2,3,11,9,4,38,84,52,7,125,523,766,365,10,364,2676,7096,7775,3006,
%T A320264 16,1041,12435,52955,100455,87261,28357,22,2838,54034,348696,1020805,
%U A320264 1497038,1074766,301064,32,7645,225417,2120284,8995801,19823964,23605043,14423564,3549177
%N A320264 Number T(n,k) of proper multisets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the multiset; triangle T(n,k), n>=2, 1<=k<=n-1, read by rows.
%H A320264 Alois P. Heinz, <a href="/A320264/b320264.txt">Rows n = 2..150</a>
%F A320264 T(n,k) = A257740(n,k) - A319501(n,k).
%e A320264 T(2,1) = 1: {a,a}.
%e A320264 T(3,2) = 2: {a,a,b}, {a,b,b}.
%e A320264 T(4,3) = 9: {a,a,b,c}, {a,a,bc}, {a,a,cb}, {b,b,a,c}, {b,b,ac}, {b,b,ca}, {c,c,a,b}, {c,c,ab}, {c,c,ba}.
%e A320264 Triangle T(n,k) begins:
%e A320264   .
%e A320264   .   .
%e A320264   .   1,    .
%e A320264   .   1,    2,     .
%e A320264   .   3,   11,     9,      .
%e A320264   .   4,   38,    84,     52,       .
%e A320264   .   7,  125,   523,    766,     365,       .
%e A320264   .  10,  364,  2676,   7096,    7775,    3006,       .
%e A320264   .  16, 1041, 12435,  52955,  100455,   87261,   28357,      .
%e A320264   .  22, 2838, 54034, 348696, 1020805, 1497038, 1074766, 301064,   .
%p A320264 h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A320264       add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
%p A320264     end:
%p A320264 g:= proc(n, k) option remember; `if`(n=0, 1, add(add(
%p A320264       d*k^d, d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
%p A320264     end:
%p A320264 T:= (n, k)-> add((-1)^i*(g(n, k-i)-h(n$2, k-i))*binomial(k, i), i=0..k):
%p A320264 seq(seq(T(n, k), k=1..n-1), n=2..12);
%t A320264 h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i<1, 0,
%t A320264      Sum[h[n - i*j, i-1, k]*Binomial[k^i, j], {j, 0, n/i}]]];
%t A320264 g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[
%t A320264      d*k^d, {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n];
%t A320264 T[n_, k_] := Sum[(-1)^i*(g[n, k-i]-h[n, n, k-i])*Binomial[k, i], {i, 0, k}];
%t A320264 Table[Table[T[n, k], {k, 1, n - 1}], {n, 2, 12}] // Flatten (* _Jean-François Alcover_, Feb 13 2021, after _Alois P. Heinz_ *)
%Y A320264 Column k=1 gives A047967.
%Y A320264 Row sums give A320265.
%Y A320264 T(n+1,n) gives A006152.
%Y A320264 Cf. A257740, A319501.
%K A320264 nonn,tabl
%O A320264 2,3
%A A320264 _Alois P. Heinz_, Oct 08 2018