login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A226909 Number of placements of brackets in a monomial of degree n in an algebra with two commutative multiplications. 5
1, 2, 4, 14, 44, 164, 616, 2450, 9908, 41116, 173144, 739884, 3196344, 13944200, 61327312, 271653254, 1210772124, 5426133764, 24435934568, 110524288836, 501864708968, 2286937749496, 10454921456688, 47936304101860, 220383617137704, 1015714229399256 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This sequence generalizes the Wedderburn-Etherington numbers (A001190) to the case of two different types of brackets, such as square brackets [-.-] and curly brackets {-,-}.
Also number of N-free graphs [Cameron]. - Michael Somos, Apr 18 2014
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence as M1302, except that, copying Cameron's error, 14 is missing).
LINKS
M. Bremner, S. Madariaga, Lie and Jordan products in interchange algebras, arXiv preprint arXiv:1408.3069, 2014.
Murray Bremner, Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. See p. 166, but note that 14 is missing. - Michael Somos, Apr 18 2014
FORMULA
G.f. A(x) satisfies A(x) = x + A(x^2) + A(x)^2. - Michael Somos, Jun 13 2014
a(n) ~ c * d^n / n^(3/2), where d = 4.8925511471743497508362229157295..., c = 0.155553379207933493345508839... . - Vaclav Kotesovec, Sep 07 2014
EXAMPLE
For n = 4 the 14 different bracketings are as follows:
[1[2[34]]], {1[2[34]]}, [1{2[34]}], {1{2[34]}}, [1[2{34}]], {1[2{34}]}, [1{2{34}}], {1{2{34}}}, [[12][34]], {[12][34]}, [[12]{34}], {[12]{34}}, [{12}{34}], {{12}{34}}.
G.f. = x + 2*x^2 + 4*x^3 + 14*x^4 + 44*x^5 + 164*x^6 + 616*x^7 + ...
MAPLE
BBcount := table():
BBcount[ 1 ] := 1:
for n from 2 to 10 do
BBcount[ n ] := 0:
for i to floor((n-1)/2) do
BBcount[n] := BBcount[n] + 2*BBcount[i]*BBcount[n-i]
od:
if n mod 2 = 0 then
BBcount[n] := BBcount[n] + 2*binomial(BBcount[n/2]+1, 2)
fi:
print( n, BBcount[ n ] )
od:
MATHEMATICA
max = 26; Clear[BBcount]; BBcount[1] = 1; For[n = 2, n <= max, n++, BBcount[n] = 0; For[i = 1, i <= Floor[(n-1)/2], i++, BBcount[n] = BBcount[n] + 2*BBcount[i]*BBcount[n-i]]; If[EvenQ[n], BBcount[n] = BBcount[n] + 2*Binomial[BBcount[n/2]+1, 2]]]; Array[BBcount, max] (* Jean-François Alcover, Mar 24 2014, translated from Maple *)
PROG
(PARI) {a(n) = local(A); if( n<2, n>0, A = x + O(x^2); for(k=2, n, A = x + A^2 + subst(A, x, x^2)); polcoeff(A, n))}; /* Michael Somos, Jun 13 2014 */
(PARI) {a(n) = if( n<2, n>0, 2 * sum(k=1, (n-1)\2, a(k) * a(n-k)) + if( n%2==0, 2 * binomial( a(n/2) + 1, 2)))}; /* Michael Somos, Jun 13 2014 */
CROSSREFS
Cf. Wedderburn-Etherington numbers (A001190), A241555.
Sequence in context: A128750 A047152 A007866 * A121751 A327644 A151355
KEYWORD
nice,nonn
AUTHOR
Murray R. Bremner, Jun 21 2013
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)