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A026641 Number of nodes of even outdegree (including leaves) in all ordered trees with n edges. 42
1, 1, 4, 13, 46, 166, 610, 2269, 8518, 32206, 122464, 467842, 1794196, 6903352, 26635774, 103020253, 399300166, 1550554582, 6031074184, 23493410758, 91638191236, 357874310212, 1399137067684, 5475504511858, 21447950506396 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of lattice paths from (0,0) to (n,n) using steps (1,0),(0,2),(1,1). - Joerg Arndt, Jun 30 2011

From Emeric Deutsch, Jan 25 2004:  (Start)

Let B = 1/sqrt(1-4*z) = g.f. for central binomial coeffs (A000984); F = (1-sqrt(1-4*z))/(z*(3-sqrt(1-4*z))) = g.f. for (A000957).

B = 1 + 2z + 6z^2 + 20z^3 + ... gives the number of nodes in all ordered trees with 0,1,2,3,... edges. On p. 288 of the Deutsch-Shapiro paper one finds that z*B*F=z + 2z^2 + 7z^3 + 24z^4 + ... gives the number of nodes of odd outdegree in all ordered trees with 1,2,3,... edges (cf. A014300).

Consequently, B - z*B*F = 2/(3*sqrt(1-4*z)-1+4*z) = 1 + z + 4z^2 + 13z^3 + 46z^4 + ... gives the total number of nodes of even degree in all ordered trees with 0,1,2,3,4,... edges.  (End)

Main diagonal of the following array: first column is filled with 1's, first row is filled alternatively with 1's or 0's: m(i,j)=m(i-1,j)+m(i,j-1): 1 0 1 0 1 ... / 1 1 2 2 3 ... / 1 2 4 6 9 ... / 1 3 7 13 22 ... / 1 4 11 24 46 ... - Benoit Cloitre, Aug 05 2002

The Hankel transform of [1,1,4,13,46,166,610,2269,...] is 3^n . - Philippe Deléham, Mar 08 2007

Second binomial transform of A127361 . - Philippe Deléham, Mar 14 2007

Starting with offset 1, generated from iterates of M * [1,1,1,...]; where M = a tridiagonal matrix with (0,2,2,2,...) in the main diagonal and (1,1,1,...) in the super and subdiagonals. - Gary W. Adamson, Jan 04 2009

Equals left border of triangle A158815. - Gary W. Adamson, Mar 27 2009

Equals the INVERTi transform of A101850: (1, 2, 7, 26, 100,...). - Gary W. Adamson, Jan 10 2012

a(n) = A035317(2*n-1,n) for n > 0. - Reinhard Zumkeller, Jul 19 2012

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Filippo Disanto, Andrea Frosini and Simone Rinaldi, Renzo Pinzani, The Combinatorics of Convex Permutominoes, Southeast Asian Bulletin of Mathematics (2008) 32: 883-912.

FORMULA

G.f. is logarithmic derivative of the generating function for the Catalan numbers A000108. So this sequence might be called the "log-Catalan" numbers. - Murray R. Bremner, Jan 25 2004

a(n) = sum(binomial(k+n, n-k)*binomial(n-k, k), k=0..floor(n/2)). - Detlef Pauly (dettodet(AT)yahoo.de), Nov 15 2001

G.f.: 2/(3*sqrt(1-4*z)-1+4*z). - Emeric Deutsch, Jul 09 2002

a(n) = (-1)^n*sum(k=0, n, (-1)^k*C(n+k, k)). - Benoit Cloitre, Aug 20 2002

a(n) = sum(binomial(2n-2j-1, n-1), j=0..floor(n/2)). - Emeric Deutsch, Jan 28 2004

A Catalan transform of the Jacobsthal numbers A001045(n+1) under the mapping G(x)-> G(xc(x)), c(x) the g.f. of A001008. The inverse mapping is H(x)->H(x(1-x)). a(n)=sum{k=0..n, (k/(2*n-k))*binomial(2*n-k, n-k)*A001045(k+1)}. - Paul Barry, Dec 18 2004

a(n) = sum{k=0..n, binomial(2n-k, k)*binomial(k, n-k)}. - Paul Barry, Jul 25 2005

a(n) = sum_{0<=k<=n-1} A126093(n,k) . - Philippe Deléham, Mar 08 2007

a(n) = (-1/2)^(n+2)+(2/3)*Sum([(4^n-k)*(binomial(2k,k))*(1/(1-2*k))*(1-(-1/8)^(n-k+1))],k=0..n). - Yalcin Aktar, Jul 06 2007

a(n) = (-1/2)^(n+2)+(3/4)*Sum(((-1/2)^(n-k))*(binomial(2k,k)),k=0..n). - Yalcin Aktar, Jul 06 2007

From Richard Choulet, Jan 22 2010: (Start)

Recurrence relations:

a(n+1) = -0.5*d(n)+1.5*binomial(2*n+1,n).

a(n+1) = a(n)+1.5*sum((1/(2*k-1))*binomial(2*k,k)*a(n+1-k),k=2..n+1).

a(n) = (2/3)*binomial(2*n,n)+(2/9)*((-2)^n/n!)*sum(prod(k-2*p),p=0..n-1)/3^k,k=0..infinity)

a(n) = sum((-1)^p*binomial(2*n-p,n),p=0..n).

a(n) = sum((1/2^(n-k+1))*binomial(n+k,k),k=0..n)^-(-0.5)^(n+1).  (End)

a(n) is the upper left term of M^n, M = an infinite square production matrix as follows:

1, 3, 0, 0, 0,...

1, 1, 1, 0, 0,...

1, 1, 1, 1, 0,...

1, 1, 1, 1, 1,...

... Also, a(n+1) is the sum of top row terms of M^n; e.g. top row of M^3 = (13, 21, 9, 3), sum = 46 = a(4), a(3) = 13. - Gary W. Adamson, Nov 22 2011

Conjecture: 2n*a(n) +(4-7n)*a(n-1) +2*(1-2n)*a(n-2)=0. - R. J. Mathar, Dec 17 2011

The conjecture is proved with the Wilf-Zeilberger (WZ) method applied to:  Sum(binomial(k+n, n-k)*binomial(n-k, k), k=0..floor(n/2)). - T. Amdeberhan, Jul 23 2012

a(n) = sum(k=0..n, binomial(n+k,k)*cos((n+k)*Pi)), which is another version of the Cloitre from above. - Arkadiusz Wesolowski, Apr 02 2012

a(n) ~ 2^(2*n+1) / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014

a(n) = binomial(2*n,n)*hypergeom([1, -n], [-2*n], -1). - Peter Luschny, May 22 2014

G.f. is the derivative of the logarithm of the g.f. for A120588. - Michael Somos, May 18 2015

a(n) = [x^n] 1/((1 - x^2)*(1 - x)^n). - Ilya Gutkovskiy, Oct 25 2017

EXAMPLE

The triangle of number of lattice paths from (0,0) to (n,k) using steps (1,0),(0,2),(1,1) begins

1;

1, 1;

1, 2, 4;

1, 3, 7, 13;

1, 4, 11, 24, 46;

1, 5, 16, 40, 86, 166;

1, 6, 22, 62, 148, 314, 610;

1, 7, 29, 91, 239, 553, 1163, 2269;

This sequence is the diagonal. [Joerg Arndt, Jul 01 2011]

G.f. = 1 + x + 4*x^2 + 13*x^3 + 46*x^4 + 166*x^5 + 610*x^6 + 2269*x^7 + ...

MAPLE

seq(add((binomial(k+n, n-k)*binomial(n-k, k)), k=0..floor(n/2)), n=1..30);

for n from 0 to 40 do d(n):=sum(binomial(2*n-k, k)*binomial(k, n-k), k=floor(n/2)..n):od:seq(b(n), n=0..40); a(0):=1:a(1):=1:for n from 1 to 40 do a(n):=(3/(2))*binomial(2*n-1, n-1)-(1/2)*d(n-1):od:seq(d(n), n=0..40); for n from 0 to 40 do a(n):=(-1/2)^(n+2)+(2/3)*sum(4^(n-k)*(binomial(2*k, k)*(1/(1-2*k))*(1-(-1/8)^(n-k+1))), k=0..n):od:seq(a(n), n=0..40); for n from 0 to 40 do a(n):=(-1/2)^(n+2)+(3/4)*sum(((-1/2)^(n-k))*(binomial(2*k, k)), k=0..n):od:seq(a(n), n=0..40); # Richard Choulet, Jan 22 2010

MATHEMATICA

f[n_] := Sum[ Binomial[n + k, k] Cos[Pi (n + k)], {k, 0, n}]; Array[f, 25, 0] (* Robert G. Wilson v, Apr 02 2012 *)

CoefficientList[Series[2/(3*Sqrt[1-4*x]-1+4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

a[ n_] := SeriesCoefficient[ D[ Log[ 1 + (1 - Sqrt[1 - 4 x]) / 2], x], {x, 0, n}]; (* Michael Somos, May 18 2015 *)

PROG

(PARI) a(n)=(-1)^n*sum(k=0, n, (-1)^k*binomial(n+k, k))

(PARI) /* same as in A092566 but use */

steps=[[1, 0], [0, 2], [1, 1]]; /* Joerg Arndt, Jun 30 2011 */

CROSSREFS

Cf. A101850, A120588, A158815.

Sequence in context: A047154 A180144 A149434 * A149435 A149436 A087440

Adjacent sequences:  A026638 A026639 A026640 * A026642 A026643 A026644

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified February 17 20:30 EST 2018. Contains 299297 sequences. (Running on oeis4.)