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A002596
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Numerators in expansion of sqrt(1+x). Absolute values give numerators in expansion of sqrt(1-x).
(Formerly M3768 N1538)
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17
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1, 1, -1, 1, -5, 7, -21, 33, -429, 715, -2431, 4199, -29393, 52003, -185725, 334305, -9694845, 17678835, -64822395, 119409675, -883631595, 1641030105, -6116566755, 11435320455, -171529806825, 322476036831, -1215486600363, 2295919134019
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OFFSET
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0,5
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COMMENTS
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Also, absolute values are numerators of (2n-3)!!/n! or the odd part of the (n-1)-th Catalan number.
The sum of the coefficients of the expansion of sqrt(1+x) is sqrt(2) (easy). Observation: The sum of the squares of the coefficients is 4/Pi.
Observation/conjecture: If a term of this sequence is divisible by a prime p, then that term is in a block of exactly (p^k-3)/2 consecutive terms all of which are divisible by p. Furthermore, if a(n) is the term preceding such a block then a(p*n-(p-1)/2) also precedes a block of (p^(k+1)-3)/2 terms all divisible by p.
E.g., a(4)=-5 is divisible by 5 and is in a block of (5^1 - 3)/2 = 1 consecutive terms that are all divisible by 5. Then a(5*3 - (5-1)/2) = a(13) = 52003 precedes a block of exactly (5^2 - 3)/2 = 11 terms all divisible by 5.
(End)
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REFERENCES
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B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 513, Eq. (7.281).
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 88.
Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994): 72.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n+2) = C(n+1)/2^k(n+1), n >= 0; where C(n) = A000108(n), k(n) = A048881(n).
a(n) = (-1)^n*numerator((1/(1-2*n))*binomial(2*n,n)/(4^n)).
(1+x)^(1/2) = Sum_{n>=0} (1/(1-2*n))*binomial(2*n,n)/(4^n)*(-x)^n.
(1-x)^(1/2) = Sum_{n>=0} (1/(1-2*n))*binomial(2*n,n)/(4^n)*(x)^n. (End)
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EXAMPLE
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sqrt(1+x) = 1 + (1/2)*x - (1/8)*x^2 + (1/16)*x^3 - (5/128)*x^4 + (7/256)*x^5 - (21/1024)*x^6 + (33/2048)*x^7 + ...
Coefficients are 1, 1/2, -1/8, 1/16, -5/128, 7/256, -21/1024, 33/2048, -429/32768, 715/65536, -2431/262144, 4199/524288, -29393/4194304, 52003/8388608, ...
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MAPLE
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MATHEMATICA
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1+InverseSeries[Series[2^p*y+y^2/2^q, {y, 0, 24}], x] (* p, q positive integers, then a(n)=numerator(y(n)). - Len Smiley, Apr 13 2000 *)
Numerator[CoefficientList[Series[Sqrt[1+x], {x, 0, 30}], x]] (* Harvey P. Dale, Oct 22 2011 *)
Table[Numerator[Product[(3 - 2 k)/(2 k) , {k, j}]], {j, 0, 30}] (* Dimitri Papadopoulos, Oct 22 2016 *)
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PROG
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(PARI) x = 'x + O('x^40); apply(x->numerator(x), Vec(sqrt(1+x))) \\ Michel Marcus, Jan 14 2016
(Magma) [(-1)^n*Numerator((1/(1-2*n))*Binomial(2*n, n)/(4^n)): n in [0..30]]; // Vincenzo Librandi, Jan 14 2016
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CROSSREFS
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Absolute values are essentially A098597.
Cf. A161198 = triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n. (End)
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KEYWORD
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easy,nice,frac,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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