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

0,2

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

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

Denominators are A056982.

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

Sequence in context: A059696 A072216 A176608 * A167735 A221517 A264247

Adjacent sequences: A277230 A277231 A277232 * A277234 A277235 A277236

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Nov 12 2016

Status

approved