A277233 Numerators of the partial sums of the squares of the expansion coefficients of 1/sqrt(1-x). Also the numerators of the Landau constants.

1, 5, 89, 381, 25609, 106405, 1755841, 7207405, 1886504905, 7693763645, 125233642041, 508710104205, 33014475398641, 133748096600189, 2165115508033649, 8754452051708621, 9054883309760265929, 36559890613417481741, 590105629859261338481, 2379942639329101454549

Offset: 0

Author

Wolfdieter Lang, Nov 12 2016

Keywords

• nonn
• frac
• easy

Comments

This is the instance m=1/2 of the partial sums r(m,n) = Sum_{k=0..n} (risefac(m,k)/ k!)^2, where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1.

The limit n -> oo does not exist. It would be hypergeometric([1/2,1/2],[1],z -> 1), which diverges.

The partial sums of the cubes converge for |m| < 2/3. See Morley's series under A277232 (for m=1/2).

a(n)/A056982(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2, where g(n) = (2*n)!/(2^n*n!)^2 = A001790(n)/A046161(n). - Peter Luschny, Sep 27 2019

Examples

			The rationals r(n) begin: 1, 5/4, 89/64, 381/256, 25609/16384, 106405/65536, 1755841/1048576, 7207405/4194304, 1886504905/1073741824, 7693763645/4294967296, ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..831
• Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe, Arch. Math. Phys. 21 (1913), 42-50. [Accessible in the USA through the Hathi Trust Digital Library.]
• Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe (Zweite Abhandlung), Arch. Math. Phys. 21 (1913), 250-255. [Accessible in the USA through the Hathi Trust Digital Library.]
• Cristinel Mortici, Sharp bounds of the Landau constants, Math. Comp. 80 (2011), pp. 1011-1018.
• G. N. Watson, The constants of Landau and Lebesgue, Quart. J. Math. Oxford Ser. 1:2 (1930), pp. 310-318.

Crossrefs

Denominators are A056982.

Cf. A001803/A046161, A277232/A241756, A001790/A046161, A005187.

Programs

Maple

a := n -> numer(add(binomial(-1/2, j)^2, j=0..n));
seq(a(n), n=0..19); # Peter Luschny, Sep 26 2019
# Alternatively:
G := proc(x) hypergeom([1/2,1/2], [1], x)/(1-x) end: ser := series(G(x), x, 20):
[seq(coeff(ser,x,n), n=0..19)]: numer(%); # Peter Luschny, Sep 28 2019

Mathematica

Accumulate[CoefficientList[Series[1/Sqrt[1-x],{x,0,20}],x]^2]//Numerator (* Harvey P. Dale, Feb 10 2019 *)
G[x_] := (2 EllipticK[x])/(Pi (1 - x));
CoefficientList[Series[G[x], {x, 0, 19}], x] // Numerator (* Peter Luschny, Sep 28 2019 *)

SageMath

def A277233(n):
    return sum((A001790(k)*(2^(A005187(n) - A005187(k))))^2 for k in (0..n))
print([A277233(n) for n in (0..19)]) # Peter Luschny, Sep 30 2019

Formula

a(n) = numerator(r(n)), with the fractional

r(n) = Sum_{k=0..n} (risefac(1/2,k)/k!)^2;

r(n) = Sum_{k=0..n} (binomial(-1/2,k))^2;

r(n) = Sum_{k=0..n} ((2*k-1)!!/(2*k)!!)^2.

The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.

r(n) ~ (log(n+3/4) + EulerGamma + 4*log(2))/Pi. - Peter Luschny, Sep 27 2019

Rational generating function: (2*K(x))/(Pi*(1-x)) where K is the complete elliptic integral of the first kind. - Peter Luschny, Sep 28 2019

a(n) = Sum_{k=0..n}(A001790(k)*(2^(A005187(n) - A005187(k))))^2. - Peter Luschny, Sep 30 2019