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 A052750 a(n) = (2*n + 1)^(n - 1). 12
 1, 1, 5, 49, 729, 14641, 371293, 11390625, 410338673, 16983563041, 794280046581, 41426511213649, 2384185791015625, 150094635296999121, 10260628712958602189, 756943935220796320321, 59938945498865420543457 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n+1) is the number of labeled incomplete ternary trees on n vertices in which each left child has a larger label than its parent. - Brian Drake, Jul 28 2008 Put a(0) = 1. For n > 0, let x(n,k) = 2*cos((2*k-1)*Pi/(2*n+1)), k=1..n. Define the recurrences S(n;0,x(n,k)) = 1, S(n;1,x(n,k)) = x(n,k), S(n;r,x(n,k)) = x(n,k)*S(n;r-1,x(n,k)) - S(n;r-2,x(n,k)), r > 1 an integer, k=1..n. CONJECTURE: For n > 0, a(n) = Product_{k=1..n} (Sum_{m=0..n-1} S(n;2*m,x(n,k))^2). - L. Edson Jeffery, Sep 11 2013 From Wolfdieter Lang, Dec 16 2013: (Start) Discriminants of the first difference of Chebyshev S-polynomials. The coefficient table for the first difference polynomials P(n, x) = S(n, x) - S(n-1, x), n >= 0, S(-1, x) = 0, with the Chebyshev S polynomials (see A049310), is given in A130777. For the discriminant of a polynomial in terms of the square of a determinant of a Vandermonde matrix build from the zeros of the polynomial see, e.g., A127670. For the proof that D(n) := discriminant(P(n,x)) = (2*n + 1)^(n - 1), n >= 1, use the formula given e.g., in the Rivlin reference, p. 218, Theorem 5.13, eq. (5.3), namely D(n) = (-1)^(n*(n-1)/2)*Product_{j=1..n} P'(n, x(n,j)), with the zeros x(n,j) = -2*cos(2*Pi*j/(2*n+1)) of P(n, x) (see A130777). P'(n, x(n,j)) = (2*n+1)*P(n-1, x(n,j))/(2*sin(Pi*j/(2*n+1))*2*cos(Pi*j/(2*n+1)))^2. P(n-1, x(n,j)) = (-1)^(n+j)*2*cos(Pi*j/(2*n+1)). Product_{j=1..n} 2*sin(Pi*j/(2*n+1)) = 2*n+1 (see the Oct 10 2013 formula in A005408. Product_{j=1..n} 2*cos(Pi*j/(2*n+1)) = 1, because S(2*n, 0) = (-1)^n. (End) a(n) is the number of labeled 2-trees with n+2 vertices, rooted at a given edge. - Nikos Apostolakis, Nov 30 2018 a(n) is also the number of 2-trees with n labeled triangles and with a distinguished oriented edge. - Nikos Apostolakis, Dec 14 2018 REFERENCES L. W. Beineke, and J. W. Moon, Several proofs of the number of labelled 2-dimensional trees, In "Proof Techniques in Graph Theory" (F. Harary editor). Academic Press, New York, 1969, pp. 11-20. Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. LINKS G. C. Greubel, Table of n, a(n) for n = 0..350 T. Fowler, I. Gessel, G. Labelle, P. Leroux, The specificiation of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, eq. (9) INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 706 Henri Muehle, Philippe Nadeau, A Poset Structure on the Alternating Group Generated by 3-Cycles, arXiv:1803.00540 [math.CO], 2018. J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. FORMULA E.g.f.: exp(-1/2*W(-2*x)), where W is Lambert's W function. E.g.f. satisfies: A(x) = sqrt(1 + 2*Sum_{n>=1} x^(2*n-1)/(2*n-1)! * A(x)^(4*n-1)). - Paul D. Hanna, Sep 07 2012 E.g.f. satisfies: A(x) = 1/A(-x*A(x)^4). - Paul D. Hanna, Sep 07 2012 a(n) = discriminant of P(n,x) = S(n,x) - S(n-1,x), n >= 1., with the Chebyshev S polynomials from A049310. For the proof see the comment above. a(n) is also the discriminant of S(n,x) + S(n-1,x) = (-1)^n*(S(n,-x) - S(n-1,-x)). - Wolfdieter Lang, Dec 16 2013 From Peter Bala, Dec 19 2013: (Start) The e.g.f. A(x) = 1 + x + 5*x^2/2! + 49*x^3/3! + 729*x^4/4! + ... satisfies: 1) A(x*exp(-2*x)) = exp(x) = 1/A(-x*exp(2*x)); 2) A^2(x) = 1/x*series reversion(x*exp(-2*x)); 3) A(x^2) = 1/x*series reversion(x*exp(-x^2)); 4) A(x) = exp(x*A(x)^2). (End) E.g.f.: sqrt(-LambertW(-2*x)/(2*x)). - Vaclav Kotesovec, Dec 07 2014 Related to A001705 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^2*log(1 + x) ) = series reversion(x - 5*x^2/2! + 26*x^3/3! - 154*x^4/4! + ...). Cf. A000272, A052752, A052774, A052782. - Peter Bala, Jun 15 2016 EXAMPLE Discriminant: n=4: P(4, x) = 1 + 2*x - 3*x^2 - x^3 + x^4 with the zeros x = -2*cos((2/9)*Pi), x = -2*cos((4/9)*Pi), x = 1, x = 2*cos((1/9)*Pi). D(4) = (Det(Vandermonde(4,[x,x,x,x]))^2 = 729 = a(4). - Wolfdieter Lang, Dec 16 2013 MAPLE spec := [S, {B=Prod(Z, S, S), S=Set(B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA max = 16; (Series[Exp[-1/2*ProductLog[-2*x]], {x, 0, max}] // CoefficientList[#, x] & ) * Range[0, max]! (* Jean-François Alcover, Jun 20 2013 *) PROG (PARI) a(n)=(2*n+1)^(n-1) \\ Charles R Greathouse IV, Nov 20 2011 (PARI) {a(n)=local(A=1+x); for(i=1, 21, A=sqrt(1+2*sum(n=1, 21, x^(2*n-1)/(2*n-1)!*A^(4*n-1))+x*O(x^n))); n!*polcoeff(A, n)} \\ Paul D. Hanna, Sep 07 2012 (MAGMA) [(2*n+1)^(n-1) : n in [0..20]]; // Wesley Ivan Hurt, Jan 20 2017 (Python) for n in range(0, 20): print((2*n + 1)**(n - 1), end=', ') # Stefano Spezia, Dec 01 2018 (GAP) List([0..20], n->(2*n+1)^(n-1)); # Muniru A Asiru, Dec 05 2018 CROSSREFS Cf. A127670, A130777, A000169, A052752, A052774, A052782, A000272. Sequence in context: A102773 A028575 A006554 * A145088 A301386 A192557 Adjacent sequences:  A052747 A052748 A052749 * A052751 A052752 A052753 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 EXTENSIONS Better description from Vladeta Jovovic, Sep 02 2003 STATUS approved

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Last modified June 18 13:35 EDT 2021. Contains 345112 sequences. (Running on oeis4.)