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A119951 Numerators of partial sums of a convergent series with value 4, involving scaled Catalan numbers A000108. 4
1, 3, 29, 65, 281, 595, 9949, 20613, 84883, 173965, 1421113, 2894229, 11762641, 23859587, 773201629, 1564082093, 6321150767, 12761711209, 102977321267, 207595672639, 836499257311, 1684433835077, 27122471168057, 54567418372945, 219485160092143, 441266239318305, 3547513302275441 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For the corresponding denominator sequence see A120069.

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).

The partial sums r(n) = Sum_{k=1..n} C(k)/2^(2*(k-1)) are rationals (written in lowest terms).

The above partial sums are equal to 4 - binomial(2n+2,n+1)/2^(2n-1). - Pieter Mostert, Oct 12 2012

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).

This sequence was essential for unraveling the structure of the row sums A160466 of the Eta triangle A160464. - Johannes W. Meijer, May 24 2009

REFERENCES

H. Meschkowski, Unendliche Reihen, 2., verb. u. erw. Aufl., Mannheim, Bibliogr. Inst., 1982, p. 32.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

W. Lang, Rationals r(n) and more.

Eric Weisstein's World of Mathematics, Catalan numbers, see eq.(10).

FORMULA

a(n) = numerator of Sum_{k=1..n} C(k)/2^(2*(k-1)).

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

EXAMPLE

Rationals r(n): [1, 3/2, 29/16, 65/32, 281/128, 595/256, 9949/4096, 20613/8192, ...]

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

A160464 is the Eta triangle.

Factor of A160466.

Sequence in context: A171409 A220953 A031912 * A296246 A257293 A221745

Adjacent sequences:  A119948 A119949 A119950 * A119952 A119953 A119954

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Jul 20 2006

STATUS

approved

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Last modified October 21 12:12 EDT 2019. Contains 328299 sequences. (Running on oeis4.)