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%I #31 Dec 04 2020 17:22:40
%S 1,3,29,65,281,595,9949,20613,84883,173965,1421113,2894229,11762641,
%T 23859587,773201629,1564082093,6321150767,12761711209,102977321267,
%U 207595672639,836499257311,1684433835077,27122471168057,54567418372945,219485160092143,441266239318305,3547513302275441
%N Numerators of partial sums of a convergent series with value 4, involving scaled Catalan numbers A000108.
%C For the corresponding denominator sequence see A120069.
%C The asymptotics for C(n)/2^(2*(k-1)) is 4/(sqrt(Pi)*k^(3/2)) (see the E. Weisstein link, also for references). The sum over the asymptotic values from k=1..infinity is (4/sqrt(Pi))*Zeta(3/2) = 5.895499840 (maple10, 10 digits).
%C The partial sums r(n) = Sum_{k=1..n} C(k)/2^(2*(k-1)) are rationals (written in lowest terms).
%C The above partial sums are equal to 4 - binomial(2n+2,n+1)/2^(2n-1). - _Pieter Mostert_, Oct 12 2012
%C The series s = Sum_{k>=1} C(k)/2^(2*(k-1)), with C(n):=A000108(n) (Catalan numbers), converges by J. L. Raabe's criterion. See the Meschkowski reference for Raabe's criterion and the example given there. The series he gives as an example can be rewritten as (1 + 4*s)/2. From the expansion of sqrt(1+x) for |x|<=1 one finds for x=-1 the value s=4 (see the W. Lang link).
%C This sequence was essential for unraveling the structure of the row sums A160466 of the Eta triangle A160464. - _Johannes W. Meijer_, May 24 2009
%D H. Meschkowski, Unendliche Reihen, 2., verb. u. erw. Aufl., Mannheim, Bibliogr. Inst., 1982, p. 32.
%H G. C. Greubel, <a href="/A119951/b119951.txt">Table of n, a(n) for n = 1..1000</a>
%H Wolfdieter Lang, <a href="/A119951/a119951.txt">Rationals r(n) and more.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Catalan numbers</a>, see eq.(10).
%F a(n) = numerator of Sum_{k=1..n} C(k)/2^(2*(k-1)).
%F a(n-1) = numerator of (1/4^n)*Sum_{i=0..n-1} (binomial(2*(i+1), i+1)*binomial(2*(n-i), n-i)), for n>=1. - _Johannes W. Meijer_, May 24 2009
%F a(n) = (2^n-(2*n+2)!/(2^(n+1)*(n+1)!^2))*gcd((n+1)!,2^(n+1)). - _Gary Detlefs_, Nov 06 2020
%e Rationals r(n): [1, 3/2, 29/16, 65/32, 281/128, 595/256, 9949/4096, 20613/8192, ...]
%t Numerator[Table[(1/4^n)*Sum[Binomial[2*(i + 1), i + 1]*Binomial[2*(n - i), n - i], {i, 0, n - 1}], {n, 1, 50}]] (* _G. C. Greubel_, Jan 31 2017 *)
%o (PARI) for(n=1,25, print1(numerator(sum(i=0,n-1, binomial(2*(i+1),i+1)* binomial(2*(n-i), n-i))/4^n), ", ")) \\ _G. C. Greubel_, Jan 31 2017
%Y A160464 is the Eta triangle.
%Y Factor of A160466.
%K nonn,easy,frac
%O 1,2
%A _Wolfdieter Lang_, Jul 20 2006
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