login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A162326 Let a(0) = a(1) = 1, and n*a(n) = 2*(-7+5*n)*a(n-1) + 9*(2-n)*a(n-2) for n >= 2. 5
1, 1, 3, 13, 71, 441, 2955, 20805, 151695, 1135345, 8671763, 67320573, 529626839, 4213228969, 33833367963, 273892683573, 2232832964895, 18314495896545, 151037687326755, 1251606057754605, 10416531069771111, 87029307323766681 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Let y = y(x) be implicitly defined by g(x,y(x)) = 0, with dg/dy not identically zero. For n >= 1, the sequence a(n) is the number of terms in the expansion of the divided difference [x0,...,xn]y in terms of bivariate divided differences of g.

(1 + 3x + 13x^2 + 71x^3 + ...) = (1 + 4x + 20x^2 + 116x^3 + ...) * 1/(1 + x + 4x^2 + 20x^3 + 116x^4 + ...); where A082298 = (1, 4, 20, 116, 740,...). - Gary W. Adamson, Nov 17 2011

The shifted sequence 1,3,13,71,.. is the binomial transform of A151374. [Georg Muntingh, Jul 19 2012].

a(n+1) is the number of Schröder paths of semilength n in which the (2,0)-steps come in 3 colors and with no peaks at level 1. [José Luis Ramírez Ramírez, Mar 31 2013]

Define an infinite triangle by T(n,0)=1 and the other cells by T(n,k)=sum_{c=0..k-1} T(n,c) + sum_{r=k..n-1} T(r,k), the sum of the cells to the left and above a cell. The column k=1 contains A000079, the column k=2 essentially A001792. Then T(n,n)=a(n) on the diagonal. - J. M. Bergot, May 22 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

G. Muntingh, Implicit Divided Differences, Little Schroeder Numbers, and Catalan Numbers, J. Integ. Seqs., Vol. 15 (2012), Article 12.6.5

FORMULA

Let E = N x N \ {(0,0), (0,1)} be a set of pairs of natural numbers. The number of terms a(n) is the coefficient of x^n*y^{n-1} of the generating function 1 - log(1 - Sum_{(s,t) in E} x^s*y^{s+t-1}) = 1 + Sum_{q >= 1} (Sum_{(s,t) in E} x^s*y^{s+t-1})^q / q.

From Georg Muntingh, Jul 19 2012: (Start)

a(n) = 2F1(1/2,1-n;2;-8), where 2F1 is the Gauss hypergeometric series.

G.f.: 5/4 - 1/4 * sqrt( (1-9*x)/(1-x) ).

Quadratic recurrence relation: a(n) = 1 + 2*sum(m=1..n-1, a(m)*a(n-m) ).

(End)

a(n) ~ 3^(2*n+1)/(16*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012

EXAMPLE

Write [0...n]y for [x0,...,xn]y and [0...s,0...t]g for [x0,...,xs;y0,...,yt]g.

For n = 1 one finds 1 term, [01]y = -[01;1]g/[0;01]g.

For n = 2 one finds 3 terms, [012]y = -[012;2]g/[0;02]g + ([01;12]g[12;2]g)/([0;02]g[1;12]g) - ([0;012]g[01;1]g[12;2]g)/([0;02]g[0;01]g[1;12]g).

a(n) = Sum_{k=0..n} (binomial(n,k)*2^(n-k-1)*binomial(2*n-2*k-2,n-k-1))/n, n>0, a(0)=1. - Vladimir Kruchinin, Mar 13 2016

MATHEMATICA

CoefficientList[Series[5/4-1/4*Sqrt[(1-9*x)/(1-x)], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PROG

(Python)

L = [1, 1]

for n in range(2, 22):

....L.append( ((-14 + 10*n)*L[-1] + (18-9*n)*L[-2])/n )

print L

# Georg Muntingh, Jul 19 2012

(PARI) a(n) = if(n<2, 1, (2*(-7+5*n)*a(n-1) + 9*(2-n)*a(n-2))/n);

vector(25, n, a(n-1)) \\ Altug Alkan, Oct 06 2015

(Maxima)

a(n):=if n=0 then 1 else sum(binomial(n, k)*2^(n-k-1)*binomial(2*n-2*k-2, n-k-1), k, 0, n)/n; /* Vladimir Kruchinin, Mar 13 2016 */

CROSSREFS

Cf. A172003, which is a generalization to bivariate implicit functions.

Cf. A003262, which is the analogous sequence for implicit derivatives, and A172004 for its generalization to bivariate implicit functions.

Cf. A082298.

Sequence in context: A277922 A182189 A198447 * A122455 A126390 A272428

Adjacent sequences:  A162323 A162324 A162325 * A162327 A162328 A162329

KEYWORD

nonn

AUTHOR

Georg Muntingh, Jul 01 2009

EXTENSIONS

Edited by Georg Muntingh, Jan 22 2010

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 20 16:25 EST 2018. Contains 299380 sequences. (Running on oeis4.)