A130549 -id:A130549 - cp's OEIS frontend

A086463 Decimal expansion of Pi^2/18.

5, 4, 8, 3, 1, 1, 3, 5, 5, 6, 1, 6, 0, 7, 5, 4, 7, 8, 8, 2, 4, 1, 3, 8, 3, 8, 8, 8, 8, 2, 0, 0, 8, 3, 9, 6, 4, 0, 6, 3, 1, 6, 6, 3, 3, 7, 3, 5, 5, 9, 9, 4, 7, 9, 2, 4, 5, 1, 8, 6, 0, 7, 6, 4, 5, 6, 6, 6, 9, 1, 5, 6, 8, 0, 1, 0, 6, 6, 9, 5, 7, 9, 4, 4, 5, 4, 2, 9, 6, 6, 8, 7, 3, 2, 5, 2, 9, 0, 1, 7, 6, 8

Offset: 0

Eric W. Weisstein, Jul 21 2003

The sequence of repeating coefficients [1,-1,-2,-1,1,2] in the sum in the formula section, is equal to the 6th column in A191898. - Mats Granvik, Mar 19 2012

			0.548311355616075478824138388882008396406316633735...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.1, p. 20.
A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, Prob. Th. and Related Fields 125, 195-224, 2003.

J. M. Borwein and R. Girgensohn, Evaluations of binomial series, Aequat. Math. 70 (2005) 25-36.
A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, arXiv:math/0206132 [math.PR], 2002.
Ji-Cai Liu, On two congruences involving Franel numbers, arXiv:2002.03650 [math.NT], 2020.
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991) 53-55.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27.
Eric Weisstein's World of Mathematics, Bootstrap Percolation.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Index entries for transcendental numbers.

Cf. A007814, A073010, A073016, A086464, A112093, A112094, A130549, A130550.

RealDigits[Pi^2/18,10,120][[1]] (* Harvey P. Dale, Aug 14 2011 *)

Pi^2/18 \\ Charles R Greathouse IV, Mar 20 2012

Sum[1/n^2/Binomial[2n,n], {n,Infinity}].
Pi^2/18 = A013661/3 = Sum[1/(i+0)^2 - 1/(i+1)^2 - 2/(i+2)^2 - 1/(i+3)^2 + 1/(i+4)^2 + 2/(i+5)^2, {i =1, 7, 13, 19, 25,.. infinity, stride of 6}]. - Mats Granvik, Mar 19 2012
Equals Sum_{k>=1} (H(k) - 2*H(2k))/((-3^k)*k). See Liu. - Michel Marcus, Feb 11 2020
Equals Sum_{k>=1} A007814(k)/k^2. - Amiram Eldar, Jul 13 2020
Equals (2/9) * Sum_{k>=0} (-1)^k*(7*k+5)*k!^3/((2*k+1)*(3*k+2)!) [Gosper 1974] - R. J. Mathar, Feb 07 2024
Continued fraction expansion: 1/(2 - 2/(13 - 48/(34 - 270/(65 - ... - 2*(2*n - 1)*n^3/(5*n^2 + 6*n + 2 - ... ))))). See A130549. - Peter Bala, Feb 16 2024

A100044 Decimal expansion of Pi^2/9.

Original entry on oeis.org

1, 0, 9, 6, 6, 2, 2, 7, 1, 1, 2, 3, 2, 1, 5, 0, 9, 5, 7, 6, 4, 8, 2, 7, 6, 7, 7, 7, 7, 6, 4, 0, 1, 6, 7, 9, 2, 8, 1, 2, 6, 3, 3, 2, 6, 7, 4, 7, 1, 1, 9, 8, 9, 5, 8, 4, 9, 0, 3, 7, 2, 1, 5, 2, 9, 1, 3, 3, 3, 8, 3, 1, 3, 6, 0, 2, 1, 3, 3, 9, 1, 5, 8, 8, 9, 0, 8, 5, 9, 3, 3, 7, 4, 6, 5, 0, 5, 8, 0, 3, 5, 3

Offset: 1

Eric W. Weisstein, Oct 31 2004

The Dirichlet L-series for the principal character mod 6 (which is A120325 shifted left) evaluated at 2. - R. J. Mathar, Jul 20 2012

Equals the asymptotic mean of the abundancy index of the numbers coprime to 6 (A007310). - Amiram Eldar, May 12 2023

			1.096622711232150957648276777764...

F. Aubonnet, D. Guinin, and B.Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
L. B. W. Jolley, Summation of Series, Dover, 1961.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Eugène-Charles Catalan, Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.
Eric Weisstein's World of Mathematics, Digit Sum.
Index entries for transcendental numbers.

Cf. A000120, A007310, A008833, A019670, A086463, A091999, A104777, A120325, A112093, A112094, A130549, A130550.

RealDigits[Pi^2/9, 10, 110][[1]] (* G. C. Greubel, Feb 17 2017 *)

default(realprecision, 110); Pi^2/9 \\ G. C. Greubel, Feb 17 2017

numerical_approx(pi^2/9, digits=120) # G. C. Greubel, Jun 02 2021

Equals 1 + (1/2)*(1/3)*(1/2) + (1/3)*(1*2)/(3*5)*(1/2)^2 + (1/4) *(1*2*3)/(3*5*7)*(1/2)^3 + .... [Jolley eq 277]
Equals 1/1^2 + 1/5^2 + 1/7^2 + 1/11^2 + 1/13^2 + 1/17^2 + .... - R. J. Mathar, Jul 20 2012
Equals 2*Sum_{n>=1} 1/(6*n*(3*n + (-1)^n - 3) - 3*(-1)^n + 5) = 2*Sum_{n>=1} 1/(2*A104777(n)). - Alexander R. Povolotsky, May 18 2014
Equals A019670^2. - Michel Marcus, May 19 2014
Equals 2*A086463 = 2*Sum_{n>=1} 1/A091999(n)^2, equivalent to the formula of 2012 above. - Alexander R. Povolotsky, May 20 2014
Equals 3F2(1,1,1; 3/2,2 ; 1/4), following from Clausen's formula of J. Reine Angew. Math 3 (1828) for squares of 2F1() as noted in A019670. - R. J. Mathar, Oct 16 2015
Equals Product_{n >= 3} prime(n)^2 / (prime(n)^2 - 1), Euler's prime product, excluding first two primes. - Fred Daniel Kline, Jun 09 2016
Equals Integral_{x=0..oo} log(x)/(x^6 - 1) dx. - Amiram Eldar, Aug 12 2020
Equals Sum_{k>=1} A000120(k) * (2*k+1)/(k^2*(k+1)^2) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021
Equals Integral_{x=0..1} log(1+x+x^2)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022
Equals Sum_{k>=1} A008833(k)/k^4. - Amiram Eldar, Jan 25 2024
Continued fraction expansion: 1/(1 - 1/(13 - 48/(34 - 270/(65 - ... - 2*(2*n-1)*n^3/((5*n^2+6*n+2) - ... ))))). See A130549. - Peter Bala, Feb 16 2024
Equals Sum_{k >= 0} 1/((k + 1)*(2*k + 1)*binomial(2*k, k)). See Catalan, Section 21, equation 30. - Peter Bala, Aug 14 2024

A130550 Denominators of partial sums for a series for 2*Zeta(2)/3 = (Pi^2)/9.

Original entry on oeis.org

1, 12, 180, 1008, 8400, 118800, 75675600, 302702400, 15437822400, 26665329600, 3226504881600, 5708431713600, 964724959598400, 964724959598400, 46628373047256000, 340112838697632000, 98292610383615648000

Offset: 1

Wolfdieter Lang, Jul 13 2007

Numerators are given in A130549.

For the rationals r(n):= 2*sum(1/(j^2*binomial(2*j,j)),j=1..n), n>=1, the van der Poorten reference and a W. Lang link see A130551.

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

Cf. A112099, A112100, A112102, A112103, A130549.

Table[2*Sum[1/(i^2*Binomial[2*i, i]), {i, 1, n}], {n, 1, 20}] // Denominator (* Vaclav Kotesovec, Mar 10 2016 *)
(2Accumulate[Table[1/(n^2 Binomial[2n,n]),{n,20}]])//Denominator (* Harvey P. Dale, Jan 27 2019 *)

a(n) = denominator(2*sum(i=1, n, 1/(i^2*binomial(2*i, i)))); \\ Michel Marcus, Mar 10 2016

a(n) = denominator(r(n)), n>=1.

Denominator of 2*Sum_{i=1..n} 1/(i^2*C(2*i,i)). - Wolfdieter Lang, Oct 07 2008, corrected by Vaclav Kotesovec, Mar 10 2016

A112099 Numerator of Sum_{i=1..n} 1/(iC(2i,i)).

Original entry on oeis.org

0, 1, 7, 3, 169, 1523, 133, 72623, 87149, 823077, 15638477, 46915441, 13834041, 224803169, 6936783521, 5587964507, 4157445593923, 12472336782289, 170187831339, 71785227258967, 153825486983593, 4905323862699739, 21820233734078929, 5695081004594650211

Offset: 0

N. J. A. Sloane, Nov 30 2005

			0, 1/2, 7/12, 3/5, 169/280, 1523/2520, 133/220, 72623/120120, 87149/144144, .... -> Pi*sqrt(3)/9.

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

Cf. A112100, A112102, A112103, A130549, A130550, A134805.

Join[{0},Accumulate[Table[1/(x Binomial[2x,x]),{x,30}]]]//Numerator (* Harvey P. Dale, May 08 2022 *)

a(n) = numerator(sum(i=1, n, 1/(i*binomial(2*i, i)))); \\ Michel Marcus, Mar 10 2016

Sum_{i >= 1} 1/(i*C(2*i, i)) = Pi*sqrt(3)/9.

Definition corrected by Wolfdieter Lang, Oct 07 2008

A134805 Denominator of Sum_{i=1..n} 1/(i^2binomial(2i,i)).

Original entry on oeis.org

1, 2, 24, 360, 2016, 16800, 237600, 151351200, 605404800, 30875644800, 53330659200, 6453009763200, 11416863427200, 1929449919196800, 1929449919196800, 93256746094512000, 680225677395264000, 196585220767231296000, 93119315100267456000, 1243794691794272409792000

Offset: 0

Wolfdieter Lang and N. J. A. Sloane, Oct 13 2008

For this sum times 2/3 see A130549/A130550 with offset 1.

			0, 1/2, 13/24, 197/360, 1105/2016, 9211/16800, 130277/237600, 82987349/151351200, ...

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

For numerators see A130549, n>=1.

seq(denom(add(1/(k^2*binomial(2*k, k)), k = 1 .. n)), n = 0 .. 19); # Peter Bala, Mar 03 2015

Join[{1},Denominator[Accumulate[Table[1/(n^2 Binomial[2n,n]),{n,20}]]]] (* Harvey P. Dale, Jun 07 2021 *)

a(n) = denominator(sum(i=1, n, 1/(i^2*binomial(2*i, i)))); \\ Michel Marcus, Mar 10 2016

Sum_{i >= 1} 1/(i^2*binomial(2*i, i)) = Pi^2/18.

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A086463 Decimal expansion of Pi^2/18.

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A100044 Decimal expansion of Pi^2/9.

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A130550 Denominators of partial sums for a series for 2*Zeta(2)/3 = (Pi^2)/9.

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A112099 Numerator of Sum_{i=1..n} 1/(i*C(2*i,i)).

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A134805 Denominator of Sum_{i=1..n} 1/(i^2*binomial(2*i,i)).

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A112099 Numerator of Sum_{i=1..n} 1/(iC(2i,i)).

A134805 Denominator of Sum_{i=1..n} 1/(i^2binomial(2i,i)).