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A025276 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5. 2
1, 0, 0, 1, 2, 4, 8, 17, 38, 88, 208, 498, 1204, 2936, 7216, 17861, 44486, 111408, 280352, 708526, 1797564, 4576472, 11688496, 29939786, 76894684, 197974480, 510864480, 1321031716, 3422685992, 8884010928, 23098674400, 60152509613, 156879556678 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Number of lattice paths from (0,0) to (n-4,0) that stay weakly in the first quadrant and such that each step is either U=(2,1), D=(2,-1), blue H=(1,0), or red h=(1,0) (n>=4). E.g. a(8)=17 because we have 16 horizontal paths of length 4 with all combinations of blue and red (1,0) steps and, in addition, UD. - Emeric Deutsch, Dec 23 2003

LINKS

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

Zhuang, Yan. A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2.

FORMULA

G.f.: [1-sqrt((1-2z)^2-4z^4)]/2. - Emeric Deutsch, Dec 23 2003

Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 4*(n-3)*a(n-2) + 4*(n-6)*a(n-4). - Vaclav Kotesovec, Jan 25 2015

MATHEMATICA

nmax = 30; aa = ConstantArray[0, nmax]; aa[[1]] = 1; aa[[2]] = 0; aa[[3]] = 0; aa[[4]] = 1; Do[aa[[n]] = Sum[aa[[k]]*aa[[n-k]], {k, 1, n-1}], {n, 5, nmax}]; aa (* Vaclav Kotesovec, Jan 25 2015 *)

PROG

(Haskell)

a025276 n = a025276_list !! (n-1)

a025276_list = 1 : 0 : 0 : 1 : f [1, 0, 0, 1] where

   f xs = x' : f (x':xs) where

     x' = sum $ zipWith (*) xs a025276_list

-- Reinhard Zumkeller, Nov 03 2011

CROSSREFS

Sequence in context: A082499 A100131 A119685 * A006461 A257300 A229202

Adjacent sequences:  A025273 A025274 A025275 * A025277 A025278 A025279

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified January 25 07:37 EST 2020. Contains 331241 sequences. (Running on oeis4.)