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A026377 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=4; also a(n)=T(2n,n-2). 3
1, 9, 58, 330, 1770, 9198, 46928, 236736, 1185645, 5909805, 29362806, 145570230, 720606705, 3563543025, 17610412600, 86989143480, 429579843435, 2121099312195, 10472653252550, 51708363376950, 255326054688320 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Number of lattice paths from (0,0) to (2n,4), using steps U=(1,1), D=(1,-1) and at even levels(zero, positive and negative) also H=(2,0). Example: a(3)=9 because we have UUUUUD, UUUUDU, UUUDUU, UUDUUU, UDUUUU, DUUUUU, HUUUU, UUHUU and UUUUH. - Emeric Deutsch, Jan 30 2004

LINKS

G. C. Greubel, Table of n, a(n) for n = 2..1000

FORMULA

From Emeric Deutsch, Jan 30 2004: (Start)

a(n) = [t^(n+2)](1+3t+t^2)^n.

a(n) = Sum_{j=ceil((n+2)/2),..,n} ( 3^(2j-n-2)*binomial(n, j)*binomial(j, n+2-j) ). (End)

From Paul Barry, Sep 20 2004: (Start)

E.g.f. : exp(3x)BesselI(2, 2x);

a(n) = Sum_{k=0..n} binomial(n, k)binomial(2k, k+2). (End)

Conjecture: n*(n+4)*a(n) -3*(n+2)*(2*n+3)*a(n-1) +5*(n+2)*(n+1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012

G.f.: (3*x-1+(1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2). - Mark van Hoeij, Apr 18 2013

a(n) ~ 5^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 07 2013

Assuming offset 0: a(n) = C(2*n+4,n)*hypergeom([-n,-n-4],[-3/2-n],-1/4). - Peter Luschny, May 09 2016

MAPLE

series( (3*x-1+(1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2), x=0, 30); # Mark van Hoeij, Apr 18 2013

MATHEMATICA

CoefficientList[Series[(3*x-1+(1-6*x+7*x^2)/Sqrt[5*x^2-6*x+1])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 07 2013 *)

PROG

(PARI) x='x+O('x^66); Vec((3*x-1+(1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2)) /* Joerg Arndt, Apr 19 2013 */

CROSSREFS

Sequence in context: A018218 A026750 A009034 * A016209 A196920 A129173

Adjacent sequences:  A026374 A026375 A026376 * A026378 A026379 A026380

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified February 16 05:10 EST 2019. Contains 320140 sequences. (Running on oeis4.)