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A300295 Denominators of (1/3)*n*(n + 2)/((1 + 2*n)*(3 + 2*n)), for n >= 0. 1
1, 15, 105, 63, 99, 429, 195, 85, 969, 133, 483, 1725, 675, 783, 2697, 1023, 385, 3885, 481, 1599, 5289, 1935, 2115, 6909, 2499, 901, 8745, 1045, 3363, 10797, 3843, 4095, 13065, 4623, 1633, 15549, 1825, 5775, 18249, 6399, 6723, 21165, 7395, 2581, 24297, 2821, 8835, 27645, 9603, 9999, 31209, 10815, 3745 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The corresponding numerators are given in A144454(n+1).

r(n) = A144454(n+1)/a(n) is the Sum_{k=0..n-1} 1/(A(k)*A(k+1)*A(k+2), with A(j) = 1 + 2*j = A005408(n) for n >= 1, and r(0) = 0. This can be  written as r(n) =  1/12 - 1/(4*A(n)*A(n+1)) = (1/3)*n*(n + 2)/(A(n)*A(n+1)). See Jolley, p. 40/41, (209), and the general remark on p. 38, (201). The value of the infinite series is therefore 1/12.

For the proof that numerator(r(n)) = A144454(n+1) one checks the formula with (mod 9) and (mod 3) given there. E.g., if n = 1 + 9*k then r(n-1) = k*(2 + 9*k)/((1 + 6*k)*(1 + 18*k)) and numerator(r(n-1)) = k*(2 + 9*k) = ((n-1)^2 - )/9 as claimed, because this ratio for r(n-1) is in lowest terms.

REFERENCES

L. B. W. Jolley, Summation of Series, Dover Publications, 2nd rev. ed., 1961, pp. 38, 40, 41.

LINKS

Table of n, a(n) for n=0..52.

FORMULA

a(n) = denominator(r(n)), with r(n) = (1/3)*n*(n + 2)/((1 + 2*n)*(3 + 2*n))), n >= 0. r(n-1) = (1/3)*(n^2 - 1)/((2*n)^2 -1), n >= 1.

G.f. for r(n) = A144454(n+1)/a(n): G(x) = (1/12)*(1 - hypergeometric([1, 2], [5/2], -x/(1-x)))/(1-x) = ((-3 + 5*x)*sqrt(x)/sqrt(1 - x) + 3*sqrt(1 - x)*(1 - x)*Arsinh(sqrt(x)/sqrt(1 - x)))/(24*x*(1 - x)*sqrt(x)/sqrt(1 - x))

  = ((-3 + 5*x)*sqrt(x/(1-x)) + 3*(1 - x)*sqrt(1 - x)*log((1 + sqrt(x))/sqrt(1 - x)))/(24*x*(1 - x)*sqrt(x/(1 - x))).

EXAMPLE

The series begins: 1/(1*3*5) + 1/(3*5*7) + 1/(5*7*9) + ...

The partial sums are r(n) = A144454(n+1)/a(n), n >= 1, and with r(0) = 0 they begin with 0/1, 1/15, 8/105, 5/63, 8/99, 35/429, 16/195, 7/85, 80/969, 11/133, 40/483, 143/1725, 56/675, 65/783, 224/2697, 85/1023, 32/385,...

PROG

(PARI) a(n) = denominator((1/3)*n*(n + 2)/((1 + 2*n)*(3 + 2*n))); \\ Michel Marcus, Mar 15 2018

CROSSREFS

Cf. A005408, A144454(n+1) (numerators).

Sequence in context: A219170 A227331 A223441 * A102791 A160892 A061550

Adjacent sequences:  A300292 A300293 A300294 * A300296 A300297 A300298

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Mar 15 2018

STATUS

approved

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Last modified May 28 04:00 EDT 2020. Contains 334671 sequences. (Running on oeis4.)