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A212303
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a(n) = n!/([(n-1)/2]!*[(n+1)/2]!) for n>0, a(0)=0, and where [ ] = floor.
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6
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0, 1, 2, 3, 12, 10, 60, 35, 280, 126, 1260, 462, 5544, 1716, 24024, 6435, 102960, 24310, 437580, 92378, 1847560, 352716, 7759752, 1352078, 32449872, 5200300, 135207800, 20058300, 561632400, 77558760, 2326762800, 300540195, 9617286240, 1166803110, 39671305740
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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E.g.f.: (1+x)*BesselI(1, 2*x).
O.g.f.: -((4*x^2-1)^(3/2)+I-(4*I)*x^2+(4*I)*x^3)/(2*x*(4*x^2-1)^(3/2)).
Recurrence: a(n) = n if n < 2 else a(n) = a(n-1)*n if n is even else a(n-1)*n*4/((n-1)*(n+1)).
a(n) = n$*floor((n+1)/2)^((-1)^n), where n$ is the swinging factorial A056040.
a(n) = Sum_{k=0..n} A189231(n, 2*k+1).
Sum_{n>=1} 1/a(n) = 2/3 + (7/27)*sqrt(3)*Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2/3 + Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022
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MAPLE
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A212303 := proc(n) if n mod 2 = 0 then n*binomial(n, iquo(n, 2))/2 else binomial(n+1, iquo(n, 2)+1)/2 fi end: seq(A212303(i), i=0..36);
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MATHEMATICA
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a[n_?EvenQ] := n*Binomial[n, n/2]/2; a[n_?OddQ] := Binomial[n+1, Quotient[n, 2]+1]/2; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Feb 05 2014 *)
nxt[{n_, a_}]:={n+1, If[OddQ[n], a(n+1), (4a(n+1))/(n(n+2))]}; Join[{0}, Transpose[ NestList[ nxt, {1, 1}, 40]][[2]]] (* Harvey P. Dale, Dec 20 2014 *)
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PROG
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(Sage)
yield 0
r, n = 1, 1
while True:
yield r
n += 1
r *= n if is_even(n) else 4*n/((n-1)*(n+1))
a = A212303(); [next(a) for i in range(36)]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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