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A115178 Expansion of c(x^2+x^3), c(x) the g.f. of A000108. 4
1, 0, 1, 1, 2, 4, 7, 15, 29, 61, 126, 266, 566, 1212, 2619, 5685, 12419, 27247, 60049, 132847, 294931, 656877, 1467258, 3286218, 7378240, 16603458, 37441990, 84599854, 191501532, 434224404, 986161959, 2243009869, 5108859821 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Diagonal sums of number triangle A117434.

a(n) = number of Motzkin n-paths (A001006) in which every flatstep (F) is followed by a downstep (D). For example, a(5)=4 counts UDUFD, UFDUD, UUDFD, UUFDD. - David Callan, Jun 07 2006

a(n) = number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps U1=(1,1), U2=(2,1) and D=(1,-1). E.g. a(6)=7 because we have U1DU1DU1D, U1U1U1DDD, U1U1DU1DD, U1DU1U1DD, U1U1DDU1D, U2DU2D and U2U2DD. - José Luis Ramírez Ramírez, May 27 2013

LINKS

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

FORMULA

a(n) = Sum_{k=0..floor(n/2)} C(k)*C(k,n-2k).

Conjecture: (n+2)*a(n) +(n+2)*a(n-1) +4*(1-n)*a(n-2) +2*(7-4*n)*a(n-3) +2*(5-2*n)*a(n-4)=0. - R. J. Mathar, Nov 15 2011

G.f. A(x) satisfies A(x) = 1 / (1 - x^2 / (1 - x / (1 - x^2 * A(x))))). - Michael Somos, May 12 2012

G.f.: (1-sqrt(1-4*z^2*(1+z)))/(2*z^2*(1+z)). - José Luis Ramírez Ramírez, May 27 2013

a(n) ~ sqrt(3 - 1/9*(-2 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3))^2) * (((-2 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3)) * (4 + (19-3*sqrt(33))^(1/3) + (19+3*sqrt(33))^(1/3)))/9)^n /(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 16 2013

EXAMPLE

1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 15*x^7 + 29*x^8 + 61*x^9 + ...

MATHEMATICA

Table[Sum[Binomial[k, n - 2*k]*CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, Feb 03 2017 *)

PROG

(PARI) {a(n) = local(A); A = O(x^0); for( k=0, n\5, A = 1 / (1 - x^2 / (1 - x / (1 - x^2 * A)))); polcoeff( A, n)} /* Michael Somos, May 12 2012 */

CROSSREFS

Cf. A007477.

Sequence in context: A136336 A244456 A232394 * A049885 A129682 A129981

Adjacent sequences:  A115175 A115176 A115177 * A115179 A115180 A115181

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 14 2006

STATUS

approved

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Last modified December 11 21:15 EST 2017. Contains 295919 sequences.