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

1, 3, 7, 51, 109, 415, 863, 13379, 27473, 107461, 219121, 1723575, 3499153, 13810887, 27956079, 884899683, 1787478201, 7085090409, 14289590493, 113433092349, 228507214803, 907912292457, 1827259905369

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

Denominators are given under A120777.

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... .

Examples

			Rationals r(n): [1, 3/4, 7/8, 51/64, 109/128, 415/512, 863/1024, 13379/16384, 27473/32768, 107461/131072, 219121/262144, ...].
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• W. Lang, Rationals r(n) and limit.

Programs

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

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

Mathematica

r[n_] := Sum[(-1/4)^k*CatalanNumber[k], {k, 0, n}]; Numerator[Table[r[n], {n, 0, 50}]] (* G. C. Greubel, Mar 27 2018 *)

PARI

{r(n) = sum(k=0,n, (-1/4)^k*binomial(2*k,k)/(k+1))};
for(n=0,30, print1(numerator(r(n)), ", ")) \\ G. C. Greubel, Mar 27 2018

Formula

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.