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A025235 a(n) = (1/2)*s(n+2), where s = A014431. 7
1, 1, 3, 7, 21, 61, 191, 603, 1961, 6457, 21595, 72975, 249085, 857013, 2970007, 10356323, 36311633, 127937649, 452738867, 1608426647, 5734534629, 20511509549, 73583105007, 264687136235, 954482676217, 3449853902761, 12495597328011, 45349353908383 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in two colors: red (R) and green (G) (i.e. Motzkin paths with the up steps in two colors). E.g. a(3)=7 because we have HHH, HRD, HGD, RDH, GDH, RHD and GHD. - Emeric Deutsch, Dec 25 2003

Equals inverse binomial transform of A071356: (1, 2, 6, 20, 72,...). - Gary W. Adamson, Sep 03 2010

a(n) is the number of increasing unary-binary trees with associated permutation that avoids 231. For more information about increasing unary-binary trees with an associated permutation, see A245888. - Manda Riehl, Aug 07 2014

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

S. Capparelli, A. Del Fra, Dyck Paths, Motzkin Paths, and the Binomial Transform, Journal of Integer Sequences, 18 (2015), #15.8.5.

Xiang-Ke Chang, X.-B. Hu, H. Lei, Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.

M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.

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.

L. W. Shapiro, C. J. Wang, A bijection between 3-Motzkin paths and Schroder paths with no peak at odd height, JIS 12 (2009) 09.3.2

FORMULA

a(n) = Sum_{k=0,..,n} 2^(k-1)*binomial(n+1, k)*binomial(n-k+1, k-1)/(n+1 ). - Len Smiley

G.f.: (1 - x - sqrt(1 - 2*x - 7*x^2)) / (4*x^2). - Michael Somos, Jun 08 2000.

G.f. (for offset 1) is series reversion of x / (1 + x + 2*x^2). - Michael Somos, Jul 12 2003.

a(n) = sum{k=0..n, binomial(n, k)2^(k/2)C(k/2)(1+(-1)^k)/2}, C(n)=A000108(n). - Paul Barry, Dec 22 2003

E.g.f.: exp(x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic, Mar 31 2004

a(n) is the leftmost term in the top row of M^n, M is an infinite square production matrix as follows:

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

2, 0, 1, 0, 0, 0,...

2, 2, 0, 1, 0, 0,...

2, 2, 2, 0, 1, 0,...

2, 2, 2, 2, 0, 1,...

2, 2, 2, 2, 2, 0,...

2, 2, 2, 2, 2, 2,...

... - Gary W. Adamson, Feb 21 2012

a(n) ~ (1+2*sqrt(2))^(n+3/2)/(2*sqrt(Pi)*2^(3/4)*n^(3/2)). - Vaclav Kotesovec, Sep 29 2012

Recurrence: (n+2)*a(n)=(2*n+1)*a(n-1)+7*(n-1)*a(n-2). - Vaclav Kotesovec, Sep 29 2012

a(n) = hypergeom([-n/2, (1-n)/2], [2], 8). - Peter Luschny, May 28 2014

G.f.: 1/(1 - x - 2*x^2/(1 - x - 2*x^2/(1 - x - 2*x^2/(1 - x - 2*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017

EXAMPLE

x + x^2 + 3*x^3 + 7*x^4 + 21*x^5 + 61*x^6 + 191*x^7 + 603*x^8 + 1961*x^9 + ...

a(4) = 21 since the top row of M^4 = (21, 11, 7, 1, 1)

MATHEMATICA

Join[{1}, Table[Sum[2^(k - 1)*Binomial[n + 1, k]*Binomial[n - k + 1, k - 1]/(n + 1), {k, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Jan 27 2017 *)

a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 2, 8];

Table[a[n], {n, 0, 27}] (* Peter Luschny, Mar 18 2018 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( serreverse( x / (1 + x + 2*x^2 + x * O(x^n))), n+1))} /* Michael Somos, Jul 12 2003 */

(PARI) {a(n) = if( n<0, 0, polcoeff( (1 - x - sqrt(1 - 2*x -7*x^2 + x^3 * O(x^n)) ) / 4, n+2))} /* Michael Somos, Mar 31 2007 */

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * simplify( polcoeff( exp(x + A) * besseli(1, 2*x * quadgen(8) + A), n)))} /* Michael Somos, Mar 31 2007 */

CROSSREFS

Cf. A071356, A001003, A068764, A217275.

Cf. A243894 is the odd indexes of a(n)

Sequence in context: A122983 A005355 A182399 * A129366 A270049 A166358

Adjacent sequences:  A025232 A025233 A025234 * A025236 A025237 A025238

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified August 22 03:39 EDT 2018. Contains 313964 sequences. (Running on oeis4.)