cp's OEIS Frontend

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.

%I A277233 #44 Sep 30 2019 04:50:22
%S A277233 1,5,89,381,25609,106405,1755841,7207405,1886504905,7693763645,
%T A277233 125233642041,508710104205,33014475398641,133748096600189,
%U A277233 2165115508033649,8754452051708621,9054883309760265929,36559890613417481741,590105629859261338481,2379942639329101454549
%N 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.
%C A277233 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.
%C A277233 The limit n -> oo does not exist. It would be hypergeometric([1/2,1/2],[1],z -> 1), which diverges.
%C A277233 The partial sums of the cubes converge for |m| < 2/3. See Morley's series under A277232 (for m=1/2).
%C A277233 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
%H A277233 Seiichi Manyama, <a href="/A277233/b277233.txt">Table of n, a(n) for n = 0..831</a>
%H A277233 Edmund Landau, <a href="https://babel.hathitrust.org/cgi/pt?id=uc1.a0002839165&amp;view=1up&amp;seq=56">Abschätzung der Koeffzientensumme einer Potenzreihe</a>, Arch. Math. Phys. 21 (1913), 42-50. [Accessible in the USA through the <a href="https://www.hathitrust.org/accessibility">Hathi Trust Digital Library</a>.]
%H A277233 Edmund Landau, <a href="https://babel.hathitrust.org/cgi/pt?id=uc1.a0002839165&amp;view=1up&amp;seq=264">Abschätzung der Koeffzientensumme einer Potenzreihe (Zweite Abhandlung)</a>, Arch. Math. Phys. 21 (1913), 250-255.  [Accessible in the USA through the <a href="https://www.hathitrust.org/accessibility">Hathi Trust Digital Library</a>.]
%H A277233 Cristinel Mortici, <a href="http://dx.doi.org/10.1090/S0025-5718-2010-02428-7">Sharp bounds of the Landau constants</a>, Math. Comp. 80 (2011), pp. 1011-1018.
%H A277233 G. N. Watson, <a href="https://doi.org/10.1093/qmath/os-1.1.310">The constants of Landau and Lebesgue</a>, Quart. J. Math. Oxford Ser. 1:2 (1930), pp. 310-318.
%F A277233 a(n) = numerator(r(n)), with the fractional
%F A277233 r(n) = Sum_{k=0..n} (risefac(1/2,k)/k!)^2;
%F A277233 r(n) = Sum_{k=0..n} (binomial(-1/2,k))^2;
%F A277233 r(n) = Sum_{k=0..n} ((2*k-1)!!/(2*k)!!)^2.
%F A277233 The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
%F A277233 r(n) ~ (log(n+3/4) + EulerGamma + 4*log(2))/Pi. - _Peter Luschny_, Sep 27 2019
%F A277233 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
%F A277233 a(n) = Sum_{k=0..n}(A001790(k)*(2^(A005187(n) - A005187(k))))^2. - _Peter Luschny_, Sep 30 2019
%e A277233 The rationals r(n) begin: 1, 5/4, 89/64, 381/256, 25609/16384, 106405/65536, 1755841/1048576, 7207405/4194304, 1886504905/1073741824, 7693763645/4294967296, ...
%p A277233 a := n -> numer(add(binomial(-1/2, j)^2, j=0..n));
%p A277233 seq(a(n), n=0..19); # _Peter Luschny_, Sep 26 2019
%p A277233 # Alternatively:
%p A277233 G := proc(x) hypergeom([1/2,1/2], [1], x)/(1-x) end: ser := series(G(x), x, 20):
%p A277233 [seq(coeff(ser,x,n), n=0..19)]: numer(%); # _Peter Luschny_, Sep 28 2019
%t A277233 Accumulate[CoefficientList[Series[1/Sqrt[1-x],{x,0,20}],x]^2]//Numerator (* _Harvey P. Dale_, Feb 10 2019 *)
%t A277233 G[x_] := (2 EllipticK[x])/(Pi (1 - x));
%t A277233 CoefficientList[Series[G[x], {x, 0, 19}], x] // Numerator (* _Peter Luschny_, Sep 28 2019 *)
%o A277233 (SageMath)
%o A277233 def A277233(n):
%o A277233     return sum((A001790(k)*(2^(A005187(n) - A005187(k))))^2 for k in (0..n))
%o A277233 print([A277233(n) for n in (0..19)]) # _Peter Luschny_, Sep 30 2019
%Y A277233 Denominators are A056982.
%Y A277233 Cf. A001803/A046161, A277232/A241756, A001790/A046161, A005187.
%K A277233 nonn,frac,easy
%O A277233 0,2
%A A277233 _Wolfdieter Lang_, Nov 12 2016