A025250 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 0, 1, 1, 0.

0, 1, 1, 0, 1, 1, 1, 3, 3, 6, 11, 15, 31, 50, 85, 161, 267, 490, 883, 1548, 2863, 5127, 9307, 17116, 31021, 57123, 104963, 192699, 356643, 658034, 1218517, 2262079, 4196895, 7812028, 14549655, 27126118, 50671255, 94697293, 177220411, 332015747

Offset

1,8

Comments

Number of lattice paths from (0,0) to (n-3,0) that stay weakly in the first quadrant and such that each step is either U=(1,1),D=(2,-1), or H=(2,0). E.g. a(10)=6 because we have HHUD, HUDH, HUHD, UDHH, UHDH and UHHD. - Emeric Deutsch, Dec 23 2003

Hankel transform of a(n+2) is Somos-4 variant A050512. - Paul Barry, Jul 05 2009

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Formula

G.f.: (1 +x^2 -sqrt(1-2*x^2-4*x^3+x^4))/2. - Michael Somos, Jun 08 2000

G.f.: x^2+x^3*(1/(1-x^2))c(x^3/(1-x^2)^2) where c(x) is the g.f. of A000108. - Paul Barry, May 20 2009

a(n+2) = Sum_{k=0..n} binomial((n+k)/2, 2*k)*(1+(-1)^(n-k))*A000108(k)/2. - Paul Barry, Jul 06 2009

a(n) = Sum_{k=0..n} binomial(k+1,n-2*k-1)*binomial(n-k-2,k)/(k+1). - Vladimir Kruchinin, Nov 22 2014

G.f.: K_{k>=0} (-x^3)/(x^2-1), where K is the Gauss notation for a continued fraction. - Benedict W. J. Irwin, Oct 11 2016

a(n) ~ sqrt(1 - r^2 - r^3) * (2*r + 4*r^2 - r^3)^n / (2*sqrt(Pi)*n^(3/2)), where r = 0.51361982956383128341133963576515885989214886200017191578885... is the root of the equation 1 - 2*r^2 - 4*r^3 + r^4 = 0. - Vaclav Kotesovec, Jul 03 2021

Shifts left 3 places under the INVERT transform. - J. Conrad, Mar 08 2023

Mathematica

Rest[CoefficientList[Series[(1+x^2-Sqrt[1-2*x^2-4*x^3+x^4])/2, {x, 0, 40}], x]]  (* Harvey P. Dale, Apr 05 2011 *)
Rest@CoefficientList[Series[x^2+ContinuedFractionK[-x^3, x^2-1, {k, 0, 40}], {x, 0, 40}], x] (* Benedict W. J. Irwin, Oct 13 2016 *)

Prog

(PARI) a(n)=polcoeff((x^2-sqrt(1-2*x^2-4*x^3+x^4+x*O(x^n)))/2, n)
(Haskell)
a025250 n = a025250_list !! (n-1)
a025250_list = 0 : 1 : 1 : f 1 [1, 1, 0] where
   f k xs = x' : f (k+1) (x':xs) where
     x' = sum $ zipWith (*) a025250_list $ take k xs
-- Reinhard Zumkeller, Nov 03 2011
(Maxima)
a(n):=sum((binomial(k+1, n-2*k-1)*binomial(n-k-2, k))/(k+1), k, 0, n); /* Vladimir Kruchinin, Nov 22 2014 */
(Magma) m:=45; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!( (1 +x^2 -Sqrt(1-2*x^2-4*x^3+x^4))/2 )); // G. C. Greubel, Feb 23 2019
(Sage) a=((1+x^2 -sqrt(1-2*x^2-4*x^3+x^4))/2).series(x, 45).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 23 2019
(GAP) List([0..45], n-> Sum([0..n], k-> Binomial(k+1, n-2*k-1)*Binomial(n-k-2, k)/(k+1) )); # G. C. Greubel, Feb 23 2019

Crossrefs

Sequence in context: A169944 A110952 A339539 * A326498 A367644 A094305

Adjacent sequences: A025247 A025248 A025249 * A025251 A025252 A025253

Keyword

nonn

Author

Clark Kimberling

Status

approved