cp's OEIS Frontend

A119951 Numerators of partial sums of a convergent series with value 4, involving scaled Catalan numbers A000108.

%I A119951 #32 Feb 16 2025 08:33:01
%S A119951 1,3,29,65,281,595,9949,20613,84883,173965,1421113,2894229,11762641,
%T A119951 23859587,773201629,1564082093,6321150767,12761711209,102977321267,
%U A119951 207595672639,836499257311,1684433835077,27122471168057,54567418372945,219485160092143,441266239318305,3547513302275441
%N A119951 Numerators of partial sums of a convergent series with value 4, involving scaled Catalan numbers A000108.
%C A119951 For the corresponding denominator sequence see A120069.
%C A119951 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 A119951 The partial sums r(n) = Sum_{k=1..n} C(k)/2^(2*(k-1)) are rationals (written in lowest terms).
%C A119951 The above partial sums are equal to 4 - binomial(2n+2,n+1)/2^(2n-1). - _Pieter Mostert_, Oct 12 2012
%C A119951 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 A119951 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 A119951 H. Meschkowski, Unendliche Reihen, 2., verb. u. erw. Aufl., Mannheim, Bibliogr. Inst., 1982, p. 32.
%H A119951 G. C. Greubel, <a href="/A119951/b119951.txt">Table of n, a(n) for n = 1..1000</a>
%H A119951 Wolfdieter Lang, <a href="/A119951/a119951.txt">Rationals r(n) and more.</a>
%H A119951 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalanNumber.html">Catalan numbers</a>, see eq.(10).
%F A119951 a(n) = numerator of Sum_{k=1..n} C(k)/2^(2*(k-1)).
%F A119951 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 A119951 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 A119951 Rationals r(n): [1, 3/2, 29/16, 65/32, 281/128, 595/256, 9949/4096, 20613/8192, ...]
%t A119951 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 A119951 (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 A119951 A160464 is the Eta triangle.
%Y A119951 Factor of A160466.
%K A119951 nonn,easy,frac
%O A119951 1,2
%A A119951 _Wolfdieter Lang_, Jul 20 2006