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 A102866 Number of finite languages over a binary alphabet (set of binary words of total length n). 18
 1, 2, 5, 16, 42, 116, 310, 816, 2121, 5466, 13937, 35248, 88494, 220644, 546778, 1347344, 3302780, 8057344, 19568892, 47329264, 114025786, 273709732, 654765342, 1561257968, 3711373005, 8797021714, 20794198581, 49024480880, 115292809910, 270495295636 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000079, A034899, A256142. Column k=2 of A292804. Row sums of A208741 and of A360634. Sequence in context: A188947 A076958 A163825 * A148368 A148369 A148370 Adjacent sequences: A102863 A102864 A102865 * A102867 A102868 A102869 KEYWORD nonn AUTHOR Philippe Flajolet, Mar 01 2005 STATUS approved

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Last modified September 22 10:22 EDT 2023. Contains 365520 sequences. (Running on oeis4.)