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A052734 a(0)=0; thereafter a(n) = Catalan(n-1)*4^(n-1)*n!. 7
0, 1, 8, 192, 7680, 430080, 30965760, 2724986880, 283398635520, 34007836262400, 4625065731686400, 703009991216332800, 118105678524343910400, 21731444848479279513600, 4346288969695855902720000, 938798417454304874987520000, 217801232849398730997104640000, 54014705746650885287281950720000, 14259882317115833715842434990080000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For n>0, the number of fully-parenthesized expressions that you can form with n operands and 4 types of binary operators. - Yogy Namara (yogy.namara(AT)gmail.com), Mar 07 2010

a(n+1) is the number of square roots of any permutation in S_{16*n} whose disjoint cycle decomposition consists of 2*n cycles of length 8. - Luis Manuel Rivera Martínez, Feb 26 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 690

Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.

Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).

W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.

FORMULA

E.g.f.: 1/8-(1/8)*(1-16*x)^(1/2).

Recurrence: {a(1)=1, (8-16*n)*a(n)+a(n+1) = 0}.

a(n) = 16^n*GAMMA(n+1/2)/Pi^(1/2).

EXAMPLE

Let's say the 4 types of binary operators are +, -, *, and /. Then, with 3 operands {a, b, c}, we can form expressions such as ((b+a)/c), (a-(c-b)), (c*(b+a)), etc. There are a(3)=192 such expressions. - Yogy Namara (yogy.namara(AT)gmail.com), Mar 07 2010

MAPLE

spec := [S, {B=Prod(C, C), S=Union(B, Z), C=Union(B, S, Z)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

seq((2*n)!/n! * 4^n, n = 0..10);

MATHEMATICA

Join[{0}, Table[CatalanNumber[n-1] 4^(n-1) n!, {n, 1, 20}]] (* Vincenzo Librandi, Mar 11 2013 *)

PROG

(MAGMA) [0] cat [Catalan(n-1)*4^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

CROSSREFS

Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.

Equal to A000108 if all operands and all operators are indistinguishable.

Sequence in context: A189537 A268095 A058873 * A003435 A071303 A128406

Adjacent sequences:  A052731 A052732 A052733 * A052735 A052736 A052737

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

Entry revised by N. J. A. Sloane, Feb 04 2013 and Feb 06 2013

STATUS

approved

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Last modified December 14 07:52 EST 2018. Contains 318090 sequences. (Running on oeis4.)