A121004 Numerators of partial sums of Catalan numbers scaled by powers of 1/(5*5^2)=1/125.

1, 126, 15752, 393801, 246125639, 30765704917, 3845713114757, 480714139345054, 12017853483626636, 7511158427266652362, 938894803408331562046, 117361850426041445314536, 14670231303255180664525012, 73351156516275903322654776, 45844472822672439576659769888, 5730559102834054947082473174969

Offset

0,2

Comments

This is the third member (p=2) of the second p-family of partial sums of normalized scaled Catalan series CsnII(p):=Sum_{k>=0} C(k)/((5^k)*F(2*p+1)^(2*k)) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi) = F(2*p+1)*sqrt(5)/phi^(2*p+1), 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 second p-family are rII(p;n):=Sum_{k=0..n} C(k)/((5^k)*F(2*p+1)^(2*k)), n>=0, for p=0,1,...

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

The value of the series is lim_{n->oo} (r(n) := rII(2;n)) = 5*(18 - 11*phi) = 5*sqrt(5)/phi^5 = 1.0081306187557...

Links

Table of n, a(n) for n=0..15.

Wolfdieter Lang, Rationals r(n), limit.

Formula

a(n) = numerator(r(n)) with r(n) = rII(p=2,n) = Sum_{k=0..n} C(k)/5^(3*k) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

Example

Rationals r(n): [1, 126/125, 15752/15625, 393801/390625, 246125639/244140625, 30765704917/30517578125, ...].

Prog

(PARI) a(n) = numerator(sum(k=0, n, binomial(2*k, k)/(k+1) / 5^(3*k))); \\ Michel Marcus, Feb 28 2026

Crossrefs

Cf. A121005 (denominators).

Cf. A000032, A000045, A000108.

The second member (p=1) is A120786/A120787.

Sequence in context: A180885 A241172 A255172 * A027491 A289326 A295838

Adjacent sequences: A121001 A121002 A121003 * A121005 A121006 A121007

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Aug 16 2006

Extensions

More terms from Michel Marcus, Feb 28 2026

Status

approved