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A101785 G.f. satisfies: A(x) = 1 + x*A(x)/(1 - x^2*A(x)^2). 5
1, 1, 1, 2, 5, 12, 30, 79, 213, 584, 1628, 4600, 13138, 37871, 110043, 321978, 947813, 2805104, 8341608, 24912004, 74686460, 224694128, 678143656, 2052640752, 6229616730, 18952875247, 57792705415, 176596786934, 540679385663 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Formula may be derived using the Lagrange Inversion theorem (cf. A049124).

a(n) = number of Dyck n-paths (A000108) all of whose descents have odd length. For example, a(3) counts UUUDDD, UDUDUD. - David Callan, Jul 25 2005

The number of noncrossing partitions of [n] with all blocks of odd size. E.g.: a(4)=5 with the five partitions being 123/4, 124/3, 134/2,1/234 and 1/2/3/4. - _Louis_ Shapiro, Jan 07 2006

Number of ordered trees with n edges in which every non-leaf vertex has an odd number of children. - David Callan, Mar 30 2007

Number of valid hook configurations of permutations of [n] that avoid the patterns 312 and 321. - Colin Defant, Apr 28 2019

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Colin Defant, Motzkin intervals and valid hook configurations, arXiv preprint arXiv:1904.10451 [math.CO], 2019.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Vaclav Kotesovec, Asymptotic of subsequences of A212382

FORMULA

a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1, k)*C(n, 2*k)/(2*k+1) for n>0, with a(0)=1.

G.f.: (1/x) * Series_Reversion( x*(1-x^2)/(1+x-x^2) ).

Recurrence: 4*n*(n+1)*(91*n^2 - 379*n + 360)*a(n) = 6*n*(182*n^3 - 849*n^2 + 1075*n - 264)*a(n-1) - 2*(182*n^4 - 1122*n^3 + 2011*n^2 - 603*n - 648)*a(n-2) + 6*(n-3)*(364*n^3 - 1698*n^2 + 2267*n - 696)*a(n-3) - 5*(n-4)*(n-3)*(91*n^2 - 197*n + 72)*a(n-4). - Vaclav Kotesovec, Sep 17 2013

a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 3/4 + 1/(4*sqrt(3/(19 - 304/(4103 + 273*sqrt(273))^(1/3) + 2*(4103 + 273*sqrt(273))^(1/3)))) + 1/2*sqrt(19/6 + 76/(3*(4103 + 273*sqrt(273))^(1/3)) - 1/6*(4103 + 273*sqrt(273))^(1/3) + 63/2*sqrt(3/(19 - 304/(4103 + 273*sqrt(273))^(1/3) + 2*(4103 + 273*sqrt(273))^(1/3)))) = 3.228704951094501729... is the root of the equation 5 - 24*d + 4*d^2 - 12*d^3 + 4*d^4 = 0 and c = 0.82499074317860885542266460957609663272... is the root of the equation -125 - 3376*c^2 - 22080*c^4 - 23296*c^6 + 93184*c^8 = 0. - Vaclav Kotesovec, added Sep 17 2013, updated Jan 04 2014

G.f.: 1/(9*(3-3*x+x^2))*(x^2+27- x^2*(2*x+3)^3*(x-6)^3/(9*(3-3*x+x^2)^3*S(0) - x^2*(2*x+3)^2*(x-6)^2 )), where S(k) = 4*k+3 - x^2*(2*x^2-9*x-18)^2*(3*k+4)*(6*k+5)/( 18*(4*k+5)*(3-3*x+x^2)^3 - x^2*(2*x^2-9*x-18)^2*(3*k+5)*(6*k+7)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2013

EXAMPLE

Generated from Fibonacci polynomials (A011973) and

coefficients of odd powers of 1/(1-x):

a(1) = 1*1/1

a(2) = 1*1/1 + 0*1/3

a(3) = 1*1/1 + 1*3/3

a(4) = 1*1/1 + 2*6/3 + 0*1/5

a(5) = 1*1/1 + 3*10/3 + 1*5/5

a(6) = 1*1/1 + 4*15/3 + 3*15/5 + 0*1/7

a(7) = 1*1/1 + 5*21/3 + 6*35/5 + 1*7/7

a(8) = 1*1/1 + 6*28/3 + 10*70/5 + 4*28/7 + 0*1/9

This process is equivalent to the formula:

a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1,k)*C(n,2*k)/(2*k+1).

MATHEMATICA

Flatten[{1, Table[Sum[Binomial[n-k-1, k]*Binomial[n, 2*k]/(2*k+1), {k, 0, Floor[(n-1)/2]}], {n, 1, 20}]}] (* Vaclav Kotesovec, Sep 17 2013 *)

CoefficientList[InverseSeries[Series[x*(1-x^2)/(1+x-x^2), {x, 0, 30}], x]/x, x] (* G. C. Greubel, May 03 2019 *)

PROG

(PARI) {a(n)=if(n==0, 1, sum(k=0, (n-1)\2, binomial(n-k-1, k)*binomial(n, 2*k)/(2*k+1)))}

for(n=1, 40, print1(a(n), ", "))

(PARI) N=66; Vec(serreverse(x/(1+sum(k=1, N, x^(2*k-1)))+O(x^N))/x) /* Joerg Arndt, Aug 19 2012 */

(MAGMA) [n eq 0 select 1 else (&+[Binomial(n-k-1, k)*Binomial(n, 2*k)/(2*k+1): k in [0..Floor((n-1)/2)]]): n in [0..30]]; // G. C. Greubel, May 03 2019

(Sage) [1]+[sum(binomial(n-k-1, k)*binomial(n, 2*k)/(2*k+1) for k in (0..floor((n-1)/2))) for n in (1..30)] # G. C. Greubel, May 03 2019

CROSSREFS

Cf. A011973, A049124.

Column k=2 of A212382.

Sequence in context: A103287 A136704 A120895 * A003089 A213263 A179544

Adjacent sequences:  A101782 A101783 A101784 * A101786 A101787 A101788

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 16 2004

STATUS

approved

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Last modified July 15 14:28 EDT 2019. Contains 325031 sequences. (Running on oeis4.)