A261043 Number of multisets of nonempty words with a total of n letters over binary alphabet such that all letters occur at least once in the multiset.

0, 0, 3, 14, 49, 148, 427, 1170, 3150, 8288, 21562, 55368, 140998, 355854, 892014, 2220856, 5497483, 13533264, 33150801, 80825768, 196218139, 474423934, 1142756063, 2742781794, 6561049181, 15645058210, 37194447065, 88174246904, 208463588035, 491585765888

Offset: 0

Author

Vaclav Kotesovec, Aug 08 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Crossrefs

Column k=2 of A257740.

Cf. A000041, A034899, A247003.

Programs

Mathematica

CoefficientList[Series[Product[1/(1-x^k)^(2^k), {k, 1, 30}] - 2*Product[1/(1 - x^k), {k, 1, 30}] + 1, {x, 0, 30}], x]
(* Second program: *)
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*k^# &]*A[n - j, k], {j, 1, n}]/n];
T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz in A257740 *)

Formula

a(n) = A034899(n) - 2*A000041(n) + 1.

a(n) ~ c^2 * 2^(n-1) * exp(2*sqrt(n) - 1/2) / (sqrt(Pi) * n^(3/4)), where c = A247003 = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502...

Extensions

New name from Alois P. Heinz, Oct 07 2018