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A052750 a(n) = (2*n+1)^(n-1). E.g.f.: exp(-1/2*W(-2*x)), where W is Lambert's W function. 11
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

Comment 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(P'(n, x(n,j)),j=1..n), 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(2*sin(Pi*j/(2*n+1))^, j=1..n) = 2*n+1 (see the Oct 10 2013 formula in A005408. product(2*cos(Pi*j/(2*n+1)),j=1..n) = 1, because S(2*n, 0) = (-1)^n.

(End)

REFERENCES

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 706

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.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

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[1] = -2*cos((2/9)*Pi), x[2] = -2*cos((4/9)*Pi), x[3] = 1, x[4] = 2*cos((1/9)*Pi). D(4) = (Det(Vandermonde(4,[x[1],x[2],x[3],x[4]]))^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);

with(finance):seq(mul(cashflows([n, n, 1], 0), k=2..n), n=0..20); # Zerinvary Lajos, Dec 22 2008

MATHEMATICA

max = 16; (Series[Exp[-1/2*ProductLog[-2*x]], {x, 0, max}] // CoefficientList[#, x] & ) * Range[0, max]! (* Jean-Fran├žois 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

CROSSREFS

Cf. A127670, A130777. A000169, A052752, A052774, A052782, A000272.

Sequence in context: A102773 A028575 A006554 * A145088 A192557 A062995

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 March 25 19:30 EDT 2017. Contains 284082 sequences.