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A357580
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a(n) = ((1 + sqrt(n))^n - (1 - sqrt(n))^n)/(2*n*sqrt(n)).
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1
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1, 1, 2, 5, 16, 57, 232, 1017, 4864, 24641, 133024, 752765, 4476928, 27707513, 178613376, 1191756593, 8231124992, 58598528065, 429868937728, 3239768599221, 25073052286976, 198825601967609, 1614604933769216, 13405327061690025, 113725655719346176
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = [x^n] x/(n*(1-2*x-(n-1)*x^2)).
a(n) = Sum_{j=0..floor(n/2)} n^(j-1) * binomial(n,2*j+1).
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MAPLE
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b:= proc(n, k) option remember;
`if`(n<2, n, 2*b(n-1, k)+(k-1)*b(n-2, k))
end:
a:= n-> b(n$2)/n:
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MATHEMATICA
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Expand[Table[((1 + Sqrt[n])^n - (1 - Sqrt[n])^n)/(2*n*Sqrt[n]), {n, 1, 27}]]
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PROG
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(Python)
from sympy import simplify, sqrt
def A357580(n): return simplify(((1+sqrt(n))**n-(1-sqrt(n))**n)/(n*sqrt(n)))>>1 # Chai Wah Wu, Oct 14 2022
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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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