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A085362 a(0)=1; for n>0, a(n) = 2*5^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i). 6
1, 2, 8, 34, 150, 678, 3116, 14494, 68032, 321590, 1528776, 7301142, 35003238, 168359754, 812041860, 3926147730, 19022666310, 92338836390, 448968093320, 2186194166950, 10659569748370, 52037098259090, 254308709196660 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of bilateral Schroeder paths (i.e. lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at even (zero, positive or negative) levels. Example: a(2)=8 because we have UDUD, UUDD, UHD, UDDU and their reflections in the x-axis. First differences of A026375. - Emeric Deutsch, Jan 28 2004

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

G.f.: sqrt((1-x)/(1-5*x)).

Sum_{i=0..n} (Sum_{j=0..i} a(j)*a(i-j)) = 5^n.

a(n) = (2*(3*n-2)*a(n-1)-5*(n-2)*a(n-2)])n; a(0)=1, a(1)=2. - Emeric Deutsch, Jan 28 2004

a(n) ~ 2*5^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012

G.f.: G(0), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)* (2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013

a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(n-1,n-k). - Vladimir Kruchinin, May 30 2016

a(n) = 2*hypergeom([3/2, 1-n], [2], -4) for n>0. - Peter Luschny, Jan 30 2017

MAPLE

a := n -> `if`(n=0, 1, 2*hypergeom([3/2, 1-n], [2], -4)):

seq(simplify(a(n)), n=0..22); # Peter Luschny, Jan 30 2017

MATHEMATICA

CoefficientList[Series[Sqrt[(1-x)/(1-5x)], {x, 0, 25}], x]

PROG

(PARI) x='x+O('x^66); Vec(sqrt((1-x)/(1-5*x))) \\ Joerg Arndt, May 10 2013

CROSSREFS

Bisection of A026392.

Cf. A026375, A026387.

Sequence in context: A245090 A151829 A026387 * A150889 A150890 A150891

Adjacent sequences:  A085359 A085360 A085361 * A085363 A085364 A085365

KEYWORD

nonn,easy

AUTHOR

Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003

STATUS

approved

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Last modified November 21 13:53 EST 2017. Contains 295001 sequences.