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A057977
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GCD of consecutive central binomial coefficients: a(n) = gcd(A001405(n+1), A001405(n)).
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35
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1, 1, 1, 3, 2, 10, 5, 35, 14, 126, 42, 462, 132, 1716, 429, 6435, 1430, 24310, 4862, 92378, 16796, 352716, 58786, 1352078, 208012, 5200300, 742900, 20058300, 2674440, 77558760, 9694845, 300540195, 35357670, 1166803110, 129644790, 4537567650
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OFFSET
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0,4
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COMMENTS
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The numbers can be seen as a generalization of the Catalan numbers, extending A000984(n)/(n+1) to A056040(n)/(floor(n/2)+1). They can also be seen as composing the aerated Catalan numbers A126120 with the aerated complementary Catalan numbers A138364. (Thus the name 'extended Catalan numbers' might be apt for this sequence.) - Peter Luschny, May 03 2011
a(n) is the number of lattice paths from (0,0) to (n,0) that do not go below the x-axis and consist of steps U=(1,1), D=(1,-1) and maximally one step H=(1,0). - Alois P. Heinz, Apr 17 2013
a(n) can be computed with ballot numbers without multiplications or divisions, see Maple program. - Peter Luschny, Feb 23 2019
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LINKS
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FORMULA
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G.f.: (4*x^2+x-1+(1-x)*sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*x^2). E.g.f.: (1+1/x)*BesselI(1, 2*x). - Vladeta Jovovic, Jan 19 2004
Recurrence: a(0) = 1 and a(n) = a(n-1)*n^[n odd]*(4/(n+2))^[n even] for n > 0.
Asymptotic formula: Let [n even] = 1 if n is even, 0 otherwise. Let N := n+1+[n even]. Then a(n) ~ 2^N /((n+1)^[n even]*sqrt(Pi*(2*N+1))).
Integral representation: a(n) = (1/(2*Pi))*Int_{x=0..4}(x^(2*n-1)* ((4-x)^2/x)^cos(Pi*n))^(1/4) (End)
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+2)//U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
D-finite with recurrence: (n+2)*a(n) - n*a(n-1) + 4*(-2*n+1)*a(n-2) + 4*(n-1)*a(n-3) + 16*(n-3)*a(n-4) = 0.
D-finite with recurrence: -(n+2)*(n^2-5)*a(n) + 4*(-2*n-1)*a(n-1) + 4*(n-1)*(n^2+2*n-4)*a(n-2) = 0. (End)
Sum_{n>=0} 1/a(n) = 8/3 + 8*Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022
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EXAMPLE
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This GCD equals A001405(n) for the smaller odd number gcd(C(12,6), C(11,5)) = gcd(924,462) = 462 = C(11,5).
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MAPLE
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A057977_ogf := proc(z) b := z -> (z-1)/(2*z^2);
(2 + b(z))/sqrt(1-4*z^2) - b(z) end:
seq(coeff(series(A057977_ogf(z), z, n+3), z, n), n = 0..35);
*(4/(n+2))^modp(n+1, 2));
A057977_int := proc(n) int((x^(2*n-1)*((4-x)^2/x)^cos(Pi*n))^(1/4), x=0..4)/(2*Pi); round(evalf(%)) end:
A057977 := n -> (n!/iquo(n, 2)!^2) / (iquo(n, 2)+1):
b := proc(p, q) option remember; local S;
if p = 0 and q = 0 then return 1 fi;
if p < 0 or p > q then return 0 fi;
S := b(p-2, q) + b(p, q-2);
if type(q, odd) then S := S + b(p-1, q-1) fi;
S end:
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MATHEMATICA
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PROG
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(PARI) a(n)=if(n<0, 0, (n+n%2)!/(n\2+1)!/(n\2+n%2)!/(1+n%2))
(Sage)
x, n = 1, 1
while True:
yield x
m = n if is_odd(n) else 4/(n+2)
x *= m
n += 1
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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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