%I #17 Feb 13 2021 14:04:58

%S 1,1,2,3,11,9,4,38,84,52,7,125,523,766,365,10,364,2676,7096,7775,3006,

%T 16,1041,12435,52955,100455,87261,28357,22,2838,54034,348696,1020805,

%U 1497038,1074766,301064,32,7645,225417,2120284,8995801,19823964,23605043,14423564,3549177

%N 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 Alois P. Heinz, <a href="/A320264/b320264.txt">Rows n = 2..150</a>

%F T(n,k) = A257740(n,k) - A319501(n,k).

%e T(2,1) = 1: {a,a}.

%e T(3,2) = 2: {a,a,b}, {a,b,b}.

%e 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 Triangle T(n,k) begins:

%e .

%e . .

%e . 1, .

%e . 1, 2, .

%e . 3, 11, 9, .

%e . 4, 38, 84, 52, .

%e . 7, 125, 523, 766, 365, .

%e . 10, 364, 2676, 7096, 7775, 3006, .

%e . 16, 1041, 12435, 52955, 100455, 87261, 28357, .

%e . 22, 2838, 54034, 348696, 1020805, 1497038, 1074766, 301064, .

%p h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))

%p end:

%p g:= proc(n, k) option remember; `if`(n=0, 1, add(add(

%p d*k^d, d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)

%p end:

%p T:= (n, k)-> add((-1)^i*(g(n, k-i)-h(n$2, k-i))*binomial(k, i), i=0..k):

%p seq(seq(T(n, k), k=1..n-1), n=2..12);

%t h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i<1, 0,

%t Sum[h[n - i*j, i-1, k]*Binomial[k^i, j], {j, 0, n/i}]]];

%t g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[

%t d*k^d, {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n];

%t T[n_, k_] := Sum[(-1)^i*(g[n, k-i]-h[n, n, k-i])*Binomial[k, i], {i, 0, k}];

%t Table[Table[T[n, k], {k, 1, n - 1}], {n, 2, 12}] // Flatten (* _Jean-François Alcover_, Feb 13 2021, after _Alois P. Heinz_ *)

%Y Column k=1 gives A047967.

%Y Row sums give A320265.

%Y T(n+1,n) gives A006152.

%Y Cf. A257740, A319501.

%K nonn,tabl

%O 2,3

%A _Alois P. Heinz_, Oct 08 2018