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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, with a(1) = 1, a(2) = a(3) = 0, a(4) = 1. 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
Yan Zhuang, A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379; arXiv:1508.02793 [math.CO], 2015-2018.
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Definition improved by Bernard Schott, Jun 27 2022
STATUS
approved

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Last modified March 29 06:44 EDT 2024. Contains 371265 sequences. (Running on oeis4.)