This site is supported by donations to The OEIS Foundation. 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 Alois P. Heinz, Table of n, a(n) for n = 1..500 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]], {1[2]}, [1{2}], {1{2}}, [1[2{34}]], {1[2{34}]}, [1{2{34}}], {1{2{34}}}, [], {}, [{34}], {{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; 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 Adjacent sequences:  A226906 A226907 A226908 * A226910 A226911 A226912 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)