A121011 -id:A121011 - cp's OEIS frontend

A121010 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*8^2) = 1/320.

1, 319, 51041, 6533247, 5226597607, 1672511234219, 267601797475073, 342530300768093011, 2192193924915795299, 17537551399326362389569, 2806008223892217982335239, 1795845263291019508694523567

Offset: 0

Wolfdieter Lang, Aug 16 2006

Denominators are given under A121011.
This is the third member (p=3) of the third p-family of partial sums of normalized scaled Catalan series CsnIII(p):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..infinity) with limit F(2*p)*(-L(2*p+1) + L(2*p)*phi) = F(2*p)*sqrt(5)/phi^(2*p), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
The partial sums of the above mentioned third p-family are rIII(p;n):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..n), n>=0, for p=1,...
For more details on this p-family and the other three ones see the W. Lang link under A120996.

			Rationals r(n): [1, 319/320, 51041/51200, 6533247/6553600,
5226597607/5242880000, 1672511234219/1677721600000,...].

W. Lang: Rationals r(n), limit.

The second member is A121008/A121009.

The limit lim_{n->infinity} (r(n) := rIII(3;n)) = 8*(-29 + 18*phi) = 8*sqrt(5)/phi^6 = 0.9968943824 (maple10, 10 digits).

a(n)=numerator(r(n)) with r(n) := rIII(p=3,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*3)^(2*k)),k=0..n), with F(6)=8 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

cp's OEIS Frontend

A121010 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*8^2) = 1/320.

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