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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. 6
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 + 3*x + 13*x^2 + 71*x^3 + ...) = (1 + 4*x + 20*x^2 + 116*x^3 + ...) * 1/(1 + x + 4*x^2 + 20*x^3 + 116*x^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 - sqrt( (1-9*x)/(1-x) ))/4.

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

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

From Peter Bala, Jan 19 2020: (Start)

a(n+1) = Sum_{k = 0..n} 2^k*C(n,k)*Catalan(k).

a(n+1) = (2/Pi) * Integral_{x = -1..1} (1 + 8*x^2)^n*sqrt(1 - x^2) dx.

O.g.f.: 1 + x/(1 - x)*c(2*x/(1-x)), where c(x) is the o.g.f. for A000108. (End)

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).

MATHEMATICA

CoefficientList[Series[(5-Sqrt[(1-9*x)/(1-x)])/4, {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

(PARI) my(x='x+O('x^20)); Vec((5-sqrt((1-9*x)/(1-x)))/4) \\ G. C. Greubel, Feb 07 2019

(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 */

(MAGMA) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (5-Sqrt((1-9*x)/(1-x)))/4 )); // G. C. Greubel, Feb 07 2019

(MAGMA) a:=[1, 3]; for n in [3..21] do Append(~a, (2*(-7+5*n)*a[n-1] + 9*(2-n)*a[n-2]) div n); end for ; [1] cat a; // Marius A. Burtea, Jan 20 2020

(Sage) ((5-sqrt((1-9*x)/(1-x)))/4).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Feb 07 2019

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, A000108, A151374.

Sequence in context: A182189 A198447 A318223 * 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

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Last modified May 27 21:52 EDT 2020. Contains 334671 sequences. (Running on oeis4.)