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A071725 Expansion of (1+x^2*C^4)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108. 2
1, 1, 3, 10, 34, 117, 407, 1430, 5070, 18122, 65246, 236436, 861764, 3157325, 11622015, 42961470, 159419670, 593636670, 2217608250, 8308432140, 31212003420, 117544456770, 443690433654, 1678353186780, 6361322162444 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = number of Dyck (n+3)-paths for which the first downstep followed by an upstep (or by nothing at all) is in position 6. For example, a(2)=3 counts UUUUDdUDDD, UUUDDdUUDD, UUUDDdUDUD (the downstep in position 6 is in small type). - David Callan, Dec 09 2004

LINKS

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

Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.

FORMULA

From Paul Barry, Jun 28 2009: (Start)

E.g.f.: exp(2x)*dif(Bessel_I(1,2x)-Bessel_I(2,2x),x);

a(n) = sum{k=0..n, C(n,k)*2^(n-k)*(-1)^k*C(k+1,floor(k/2))}. (End)

Conjecture: (n+31)*(n+3)*a(n) +(n^2-180*n-219)*a(n-1) -10*(2*n-3)*(n-10)*a(n-2) = 0 . - R. J. Mathar, Nov 23 2011

a(n) ~ 3*2^(2*n+1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014

MATHEMATICA

CoefficientList[Series[(1 + x^2 ((1 - (1 - 4 x)^(1/2))/(2 x))^4) (1 - (1 - 4 x)^(1/2))/(2 x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

CROSSREFS

Essentially the same as A026016.

Sequence in context: A059738 A094832 A217778 * A026016 A109263 A136439

Adjacent sequences:  A071722 A071723 A071724 * A071726 A071727 A071728

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 06 2002

STATUS

approved

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Last modified November 21 04:33 EST 2019. Contains 329350 sequences. (Running on oeis4.)