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

2,2

Links

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

Formula

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

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

Example

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}.

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 *)

Crossrefs

Row sums of A320264.

Cf. A257741, A319518.

Sequence in context: A002398 A110065 A002816 * A144479 A074579 A060880

Adjacent sequences: A320262 A320263 A320264 * A320266 A320267 A320268

Keyword

nonn

Author

Alois P. Heinz, Oct 08 2018

Status

approved