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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.

%I A261043 #15 Jun 01 2022 14:45:28
%S A261043 0,0,3,14,49,148,427,1170,3150,8288,21562,55368,140998,355854,892014,
%T A261043 2220856,5497483,13533264,33150801,80825768,196218139,474423934,
%U A261043 1142756063,2742781794,6561049181,15645058210,37194447065,88174246904,208463588035,491585765888
%N 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.
%H A261043 Vaclav Kotesovec, <a href="/A261043/b261043.txt">Table of n, a(n) for n = 0..3000</a>
%F A261043 a(n) = A034899(n) - 2*A000041(n) + 1.
%F A261043 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...
%t A261043 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]
%t A261043 (* Second program: *)
%t A261043 A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*k^# &]*A[n - j, k], {j, 1, n}]/n];
%t A261043 T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
%t A261043 a[n_] := T[n, 2];
%t A261043 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 01 2022, after _Alois P. Heinz_ in A257740 *)
%Y A261043 Column k=2 of A257740.
%Y A261043 Cf. A000041, A034899, A247003.
%K A261043 nonn
%O A261043 0,3
%A A261043 _Vaclav Kotesovec_, Aug 08 2015
%E A261043 New name from _Alois P. Heinz_, Oct 07 2018