A120998 - OEIS

%I #8 Aug 29 2019 17:37:14

%S 1,50,2452,120153,841073,41212583,14135916101,692659889378,

%T 33940334580952,1663076394471510,81490743329120786,570435203303853900,

%U 27951324961888870816,9587304461927883432788,469777918634466290881052

%N Numerators of partial sums of Catalan numbers scaled by powers of 1/7^2 = 1/49.

%C Denominators are given under A120999.

%C This is the third member (p=2) of the first p-family of partial sums of normalized scaled Catalan series CsnI(p):=sum(C(k)/L(2*p)^(2*k),k=0..infinity) with limit L(2*p)*(F(2*p+1) - F(2*p)*phi) = L(2*p)/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).

%C The partial sums of the above mentioned first p-family are rI(p;n):=sum(C(k)/L(2*p)^(2*k),k=0..n), n>=0, for p=0,1,...

%C For more details on this p-family and the other three ones see the W. Lang link under A120996.

%C The limit lim_{n->infinity} r(n) = 7*(5 - 3* phi) = 7/phi^4 = 1.0212862362522 (maple10, 15 digits).

%H W. Lang: <a href="/A120998/a120998.txt">Rationals r(n), limit.</a>

%F a(n)=numerator(r(n)) with r(n) := rI(p=2,n) = sum(C(k)/L(4)^(2*k),k=0..n), with Lucas L(4)=7 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

%e Rationals r(n): [1, 50/49, 2452/2401, 120153/117649, 841073/823543,

%e 41212583/40353607, 14135916101/13841287201,...].

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 16 2006