A002596 Numerators in expansion of sqrt(1+x). Absolute values give numerators in expansion of sqrt(1-x).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nice
• frac
• sign

Comments

Also, absolute values are numerators of (2n-3)!!/n! or the odd part of the (n-1)-th Catalan number.

From Dimitri Papadopoulos, Oct 28 2016: (Start)

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)

Examples

			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, ...
		

References

• 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).
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:6 at page 51.

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Tom Copeland, Addendum to Elliptic Lie Triad.
• Eric Weisstein's World of Mathematics, Legendre Polynomial.

Crossrefs

Denominators are A046161.

Cf. A001795.

Equals A000265(A000108(n-1)), n>0.

Absolute values are essentially A098597.

From Johannes W. Meijer, Jun 08 2009: (Start)

Cf. A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/2)], A001803 [(1-x)^(-3/2)].

Cf. A161198 = triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n. (End)

Programs

Magma

[(-1)^n*Numerator((1/(1-2*n))*Binomial(2*n,n)/(4^n)): n in [0..30]]; // Vincenzo Librandi, Jan 14 2016

Maple

seq(numer(subs(k=1/2,expand(binomial(k,n)))),n=0..50); # James R. Buddenhagen, Aug 16 2014

Mathematica

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 *)

PARI

x = 'x + O('x^40); apply(x->numerator(x), Vec(sqrt(1+x))) \\ Michel Marcus, Jan 14 2016

Formula

a(n+2) = C(n+1)/2^k(n+1), n >= 0; where C(n) = A000108(n), k(n) = A048881(n).

From Johannes W. Meijer, Jun 08 2009: (Start)

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)

a(n) = numerator(Product_{k=1..n} (3-2*k)/(2*k)). - Dimitri Papadopoulos, Oct 22 2016

Extensions

Minor correction to definition from Johannes W. Meijer, Jun 05 2009