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

0,3

Links

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

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...

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 *)

Crossrefs

Column k=2 of A257740.

Cf. A000041, A034899, A247003.

Sequence in context: A139263 A354507 A337075 * A063025 A187917 A164304

Adjacent sequences: A261040 A261041 A261042 * A261044 A261045 A261046

Keyword

nonn

Author

Vaclav Kotesovec, Aug 08 2015

Extensions

New name from Alois P. Heinz, Oct 07 2018

Status

approved