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A120788 Numerators of partial sums of Catalan numbers scaled by powers of -1/4.

%I A120788 #20 Sep 08 2022 08:45:26
%S A120788 1,3,7,51,109,415,863,13379,27473,107461,219121,1723575,3499153,
%T A120788 13810887,27956079,884899683,1787478201,7085090409,14289590493,
%U A120788 113433092349,228507214803,907912292457,1827259905369
%N A120788 Numerators of partial sums of Catalan numbers scaled by powers of -1/4.
%C A120788 Denominators are given under A120777.
%C A120788 From the expansion of sqrt(2) = 1 + (1/2)*Sum_{k>=0} C(k)/(-4)^k, where C(n) are Catalan numbers, one has, with the partial sums, r(n), as defined in the formula section, r = limit_{n to infinity} r(n) = 2*(sqrt(2)-1) = 0.828427124... .
%H A120788 G. C. Greubel, <a href="/A120788/b120788.txt">Table of n, a(n) for n = 0..1000</a>
%H A120788 W. Lang, <a href="/A120788/a120788.txt">Rationals r(n) and limit.</a>
%F A120788 a(n) = numerator(r(n)), with the rationals r(n) := Sum_{k=0..n}((-1)^k * C(k)/4^k) with C(k) = A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
%e A120788 Rationals r(n): [1, 3/4, 7/8, 51/64, 109/128, 415/512, 863/1024, 13379/16384, 27473/32768, 107461/131072, 219121/262144, ...].
%t A120788 r[n_] := Sum[(-1/4)^k*CatalanNumber[k], {k, 0, n}]; Numerator[Table[r[n], {n, 0, 50}]] (* _G. C. Greubel_, Mar 27 2018 *)
%o A120788 (PARI) {r(n) = sum(k=0,n, (-1/4)^k*binomial(2*k,k)/(k+1))};
%o A120788 for(n=0,30, print1(numerator(r(n)), ", ")) \\ _G. C. Greubel_, Mar 27 2018
%o A120788 (Magma) [Numerator((&+[(-1/4)^k*Binomial(2*k,k)/(k+1): k in [0..n]])): n in [0..30]]; // _G. C. Greubel_, Mar 27 2018
%o A120788 (GAP) List(List([0..25], n->Sum([0..n],k->(-1/4)^k*Binomial(2*k,k)/(k+1))),NumeratorRat); # _Muniru A Asiru_, Mar 30 2018
%K A120788 nonn,easy,frac
%O A120788 0,2
%A A120788 _Wolfdieter Lang_, Jul 20 2006