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A025250
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a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-3)*a(3) for n >= 4.
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5
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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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 (deutsch(AT)duke.poly.edu), Dec 23 2003
Hankel transform of a(n+2) is Somos-4 variant A050512. [From Paul Barry (pbarry(AT)wit.ie), Jul 05 2009]
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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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. [From Paul Barry (pbarry(AT)wit.ie), May 20 2009]
a(n+2)=sum{k=0..n, C((n+k)/2,2k)(1+(-1)^(n-k))*A000108(k)/2}. [From Paul Barry (pbarry(AT)wit.ie), Jul 06 2009]
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MATHEMATICA
| Rest[CoefficientList[Series[(1+x^2-Sqrt[1-2*x^2-4*x^3+x^4])/2, {x, 0, 40}], x]] (* From Harvey P. Dale, Apr 05 2011 *)
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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
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CROSSREFS
| Sequence in context: A049926 A169944 A110952 * A094305 A057963 A112434
Adjacent sequences: A025247 A025248 A025249 * A025251 A025252 A025253
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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