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A362581
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Number of alternating permutations on [2n+1] with 1 in position n+1.
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2
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1, 2, 6, 80, 1750, 64512, 3438204, 253913088, 24687555750, 3062092267520, 471565937953396, 88298062293762048, 19753693667117055100, 5203824518733863321600, 1594426273578194363292600, 562191171748426920367226880, 226024705816530632892282399750
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = binomial(2*n,n)*A000111(n)^2:
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EXAMPLE
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a(0) = 1: 1.
a(1) = 2: 213, 312.
a(2) = 6: 23154, 24153, 25143, 34152, 35142, 45132.
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MAPLE
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b:= proc(u, o) option remember; `if`(u+o=0, 1,
add(b(o-1+j, u-j), j=1..u))
end:
a:= n-> binomial(2*n, n)*b(n, 0)^2:
seq(a(n), n=0..20);
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PROG
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(Python)
from itertools import accumulate
from math import comb
if n <= 1: return n+1
blist = (0, 1)
for _ in range(n-1):
blist = tuple(accumulate(reversed(blist), initial=0))
return blist[-1]**2*comb(n<<1, n) # Chai Wah Wu, Apr 25 2023
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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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