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A023431
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Generalized Catalan Numbers.
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15
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1, 1, 1, 2, 4, 7, 13, 26, 52, 104, 212, 438, 910, 1903, 4009, 8494, 18080, 38656, 82988, 178802, 386490, 837928, 1821664, 3970282, 8673258, 18987930, 41652382, 91539466, 201525238, 444379907, 981384125, 2170416738, 4806513660
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OFFSET
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0,4
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COMMENTS
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Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(2,-1). E.g. a(5)=7 because we have HHHHH, HHUD, HUDH, HUHD, UDHH, UHDH and UHHD. - Emeric Deutsch, Dec 25 2003
Also number of peakless Motzkin paths of length n with no double rises; in other words, Motzkin paths of length n with no UD's and no UU's, where U=(1,1) and D=(1,-1). E.g. a(5)=7 because we have HHHHH, HHUHD, HUHDH, HUHHD, UHDHH, UHHDH and UHHHD, where H=(1,0). - Emeric Deutsch, Jan 09 2004
Series reversion of g.f. A(x) is -A(-x) (if offset 1). - Michael Somos, Jul 13 2003
Number of FU_{k}-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths. This also works for U_{k}F-equivalence classes. - Sergey Kirgizov, Apr 08 2018
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LINKS
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FORMULA
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G.f.: (1 - x - sqrt((1-x)^2 - 4*x^3)) / (2*x^3) = A(x). y = x * A(x) satisfies 0 = x - y + x*y + (x*y)^2. - Michael Somos, Jul 13 2003
a(n+1) = a(n) + a(0)*a(n-2) + a(1)*a(n-3) + ... + a(n-2)*a(0). - Michael Somos, Jul 13 2003
G.f.: (1/(1-x))*c(x^3/(1-x)^2), c(x) the g.f. of A000108. - From Paul Barry, Sep 19 2008
G.f.: 1/(1-x-x^3/(1-x-x^3/(1-x-x^3/(1-x-x^3/(1-... (continued fraction).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-k, 2*k)*A000108(k). (End)
(n+3)*a(n) = (2*n+3)*a(n-1) - n*a(n-2) + 2*(2*n-3)*a(n-3). - R. J. Mathar, Nov 26 2012
0 = a(n)*(16*a(n+1) - 10*a(n+2) + 32*a(n+3) - 22*a(n+4)) + a(n+1)*(2*a(n+1) - 15*a(n+2) + 9*a(n+3) + 4*a(n+4)) + a(n+2)*(a(n+2) + 2*a(n+3) - 5*a(n+4)) + a(n+3)*(a(n+3) + a(n+4)) if n>=0. - Michael Somos, Jan 30 2014
a(n) ~ (8 + 12*r^2 + 5*r) * sqrt(r^2 - 4*r + 3) / (4 * sqrt(Pi) * n^(3/2) * r^n), where r = 0.432040800333095789... is the real root of the equation -1 + 2*r - r^2 + 4*r^3 = 0. - Vaclav Kotesovec, Jun 15 2022
a(n) = hypergeom([(1 - n)/3, (2 - n)/3, -n/3], [2, -n], 27). - Peter Luschny, Jun 15 2022
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EXAMPLE
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G.f. = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 13*x^6 + 26*x^7 + 52*x^8 + 104*x^9 + ...
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MAPLE
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a := n -> hypergeom([1/3 - n/3, 2/3 - n/3, -n/3], [2, -n], 27):
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MATHEMATICA
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a[0]=1; a[n_]:= a[n]= a[n-1] + Sum[a[k]*a[n-3-k], {k, 0, n-3}];
Table[a[n], {n, 0, 40}]
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PROG
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(PARI) {a(n) = polcoeff( (1 - x - sqrt((1-x)^2 - 4*x^3 + x^4 * O(x^n))) / 2, n+3)}; /* Michael Somos, Jul 13 2003 */
(Haskell)
a023431 n = a023431_list !! n
a023431_list = 1 : 1 : f [1, 1] where
f xs'@(x:_:xs) = y : f (y : xs') where
y = x + sum (zipWith (*) xs $ reverse $ xs')
(Magma) [(&+[Binomial(n-k, 2*k)*Catalan(k): k in [0..Floor(n/3)]]): n in [0..40]]; // G. C. Greubel, Jun 15 2022
(SageMath) [sum(binomial(n-k, 2*k)*catalan_number(k) for k in (0..(n//3))) for n in (0..40)] # G. C. Greubel, Jun 15 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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