A320265 Number of proper multisets of nonempty words with a total of n letters over n-ary alphabet such that if a letter occurs in the multiset all predecessors occur at least once.

1, 3, 23, 178, 1786, 20927, 282520, 4299263, 72750927, 1353700567, 27452623890, 602326265519, 14209892886819, 358576428141962, 9634718410829852, 274567642777650028, 8270000441627265937, 262464788618069324640, 8752908129221863491691, 305968679117675345995513

Offset: 2

Author

Alois P. Heinz, Oct 08 2018

Keywords

• nonn

Examples

			a(2) = 1: {a,a}.
a(3) = 3: {a,a,a}, {a,a,b}, {a,b,b}.
a(4) = 23: {a,a,a,a}, {a,a,aa}, {aa, aa}, {a,a,a,b}, {a,a,b,b}, {a,b,b,b}, {a,a,ab}, {a,a,ba}, {a,a,bb}, {b,b,ab}, {b,b,ba}, {b,b,aa}, {ab,ab}, {ba,ba}, {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}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..300

Crossrefs

Row sums of A320264.

Cf. A257741, A319518.

Programs

Maple

h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      d*k^d, d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> add(add((-1)^i*(g(n, k-i)-h(n$2, k-i))*
        binomial(k, i), i=0..k), k=1..n-1):
seq(a(n), n=2..25);

Mathematica

h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[h[n - i*j, i - 1, k]*Binomial[k^i, j], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[d*k^d, {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n];
T[n_, k_] := Sum[(-1)^i*(g[n, k-i]-h[n, n, k-i])*Binomial[k, i], {i, 0, k}];
a[n_] := Sum[T[n, k], {k, 1, n - 1}];
Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz in A320264 *)

Formula

a(n) = Sum_{k=1..n-1} A320264(n,k).

a(n) = A257741(n) - A319518(n).