OFFSET
0,2
COMMENTS
Analogous to A034899 (which also enumerates multisets of words)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 64
Stefan Gerhold, Counting finite languages by total word length, INTEGERS 11 (2011), #A44.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
FORMULA
G.f.: exp(Sum((-1)^(j-1)/j*(2*z^j)/(1-2*z^j), j=1..infinity)).
Asymptotics (Gerhold, 2011): a(n) ~ c * 2^(n-1)*exp(2*sqrt(n)-1/2) / (sqrt(Pi) * n^(3/4)), where c = exp( Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1)) ) = 0.6602994483152065685... . - Vaclav Kotesovec, Sep 13 2014
Weigh transform of A000079. - Alois P. Heinz, Jun 25 2018
EXAMPLE
a(2) = 5 because the sets are {a,b}, {aa}, {ab}, {ba}, {bb}.
a(3) = 16 because the sets are {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}, {aaa}, {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {bbb}.
MAPLE
series(exp(add((-1)^(j-1)/j*(2*z^j)/(1-2*z^j), j=1..40)), z, 40);
MATHEMATICA
nn = 20; p = Product[(1 + x^i)^(2^i), {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 07 2012 *)
CoefficientList[Series[E^Sum[(-1)^(k-1)/k*(2*x^k)/(1-2*x^k), {k, 1, 30}], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 13 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Flajolet, Mar 01 2005
STATUS
approved