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A124396
Denominators of partial sums of a series for 3/sqrt(5) = (3/5)*sqrt(5).
3
1, 9, 27, 729, 6561, 6561, 177147, 1594323, 4782969, 387420489, 3486784401, 10460353203, 282429536481, 2541865828329, 2541865828329, 22876792454961, 205891132094649, 617673396283947, 50031545098999707, 450283905890997363
OFFSET
0,2
COMMENTS
Denominators of sums over central binomial coefficients scaled by powers of 9.
Numerators are given by A123749.
For the rationals r(n) see the W. Lang link under A123749.
This is not 3/5 times the rational sequence A123747/A123748 which converges to sqrt(5).
LINKS
FORMULA
a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/9^k in lowest terms.
r(n) = Sum_{k=0..n} ((2*k-1)!!/((2*k)!!)*(4/9)^k, n>=0, with the double factorials A001147 and A000165.
EXAMPLE
a(3) = 729 because r(3) = 1 + 2/9 + 2/27 + 20/729 = 965/729 = A123749(3)/a(3).
MAPLE
seq(denom(add(binomial(2*k, k)/9^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019
MATHEMATICA
Table[Denominator[Sum[(k+1)*CatalanNumber[k]/9^k, {k, 0, n}]], {n, 0, 20}] (* G. C. Greubel, Dec 25 2019 *)
PROG
(PARI) a(n) = denominator(sum(k=0, n, binomial(2*k, k)/9^k)); \\ Michel Marcus, Aug 12 2019
(Magma) [Denominator(&+[(k+1)*Catalan(k)/9^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019
(Sage) [denominator(sum((k+1)*catalan_number(k)/9^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019
(GAP) List([0..20], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k, k)/9^k)) ); # G. C. Greubel, Dec 25 2019
CROSSREFS
Cf. A123749 (numerators).
Cf. A123747/A123748 partial sums for a series for sqrt(5).
Sequence in context: A320676 A317027 A020254 * A020281 A075539 A225300
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Nov 10 2006
STATUS
approved