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A026016
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a(n) = binomial(2*n-1, n) - binomial(2*n-1, n+3).
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6
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1, 3, 10, 34, 117, 407, 1430, 5070, 18122, 65246, 236436, 861764, 3157325, 11622015, 42961470, 159419670, 593636670, 2217608250, 8308432140, 31212003420, 117544456770, 443690433654, 1678353186780, 6361322162444, 24155384502452, 91882005146652
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OFFSET
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1,2
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COMMENTS
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Number of (s(0), s(1), ..., s(2n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n-1) = 3. Also a(n) = T(2n-1,n-1), where T is the array defined in A026009.
Number of integer lattice paths from (0,2) to (n-1,n+2) that do not cross the main diagonal.
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LINKS
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FORMULA
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Expansion of (1+x^1*C^3)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
(n+3)*a(n) +(-7*n-9)*a(n-1) +2*(7*n-4)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jun 20 2013
G.f.: (1-x)*( 1 -4*x +2*x^2 - (1-2*x)*sqrt(1-4*x) )/(2*x^3).
E.g.f.: -1 + (exp(2*x)/x^2)*( x*(1+x)*BesselI(0, 2*x) - (2 +x +2*x^2)*Bessel(1, 2*x) ).
a(n) = C(n) + Sum_{j=0..n-2} C(n-j-2)*(C(j+3) - 2*C(j+2)), where C(n) are the Catalan numbers. (End)
a(n) = C(n+2) -3*C(n+1) +2*C(n) = 6*((n^2+1)/((n+2)*(n+3)))*C(n). - G. C. Greubel, Mar 22 2021
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MAPLE
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a:= n-> binomial(2*n-1, n) -binomial(2*n-1, n+3): seq(a(n), n=1..27); # Zerinvary Lajos, Dec 10 2007
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MATHEMATICA
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Table[Binomial[2 n - 1, n] - Binomial[2 n - 1, n + 3], {n, 1, 40}] (* Vincenzo Librandi, Jun 21 2013 *)
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PROG
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(Magma) [Binomial(2*n-1, n) - Binomial(2*n-1, n+3): n in [1..30]]; // Vincenzo Librandi, Jun 21 2013
(Sage) [binomial(2*n-1, n) - binomial(2*n-1, n+3) for n in (2..30)] # G. C. Greubel, Mar 19 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Better description from Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001
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STATUS
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approved
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