A145434 -id:A145434 - cp's OEIS frontend

A145433 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*n/binomial(2n,n).

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

Offset: 0

R. J. Mathar, Feb 08 2009

			0.274432715277120323...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.39

Steven Finch, Central Binomial Coefficients [Cached copy, with permission of the author]

Cf. A002163, A002390, A145434.

6/25+4/125*5^(1/2)*ln(1/2+1/2*5^(1/2)) ;

RealDigits[6/25 + 4*Sqrt[5]*Log[GoldenRatio]/125, 10, 93] // First (* Jean-François Alcover, Oct 27 2014 *)
RealDigits[Hypergeometric2F1[2, 2, 3/2, -1/4]/2, 10, 93] // First (* Vaclav Kotesovec, Oct 27 2014 *)

Equals 2*(15+2*A002163*A002390)/125.

A157701 Decimal expansion of 2(14sigma+5)/625 where sigma = sqrt(5)*log(golden ratio).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 04 2009

The factor 28 in the Lehmer paper has been corrected to 14.

Equals sum_{n=1..infinity} (-1)^n*n^3/binomial(2n,n).

			0.064205801387968452355..

D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly 92, no 7 (1985) 449-457.
R. J. Mathar, Corrigenda to "Interesting series involving...", arXiv:0905.0215
Index entries for transcendental numbers

Cf. A145434, A145433.

2/625*(14*sqrt(5)*log((1+sqrt(5))/2)+5) ;

Join[{0},RealDigits[2*(14*Sqrt[5]*Log[GoldenRatio]+5)/625,10,120][[1]]] (* Harvey P. Dale, Mar 13 2015 *)

2*(14*sqrt(5)*log((sqrt(5)+1)/2)+5)/625 \\ Charles R Greathouse IV, May 15 2019

Equals 2*(14*A002163*A002390+5)/625 .

A307086 Decimal expansion of 4(5 - sqrt(5)log(phi))/25, where phi is the golden ratio (A001622).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 23 2019

Decimal expansion of the alternating sum of the reciprocals of the central binomial coefficients (A000984).

			1/1 - 1/2 + 1/6 - 1/20 + 1/70 - 1/252 + ... = 0.62783642361439838444422670681975782983017172698388...

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient
Index entries for transcendental numbers

Cf. A000984, A001622, A073016, A086465, A091682, A145433, A145434.

RealDigits[4 (5 - Sqrt[5] Log[GoldenRatio])/25, 10, 120][[1]]

4*(5 - sqrt(5)*log((sqrt(5)+1)/2))/25 \\ Charles R Greathouse IV, May 15 2019

Equals Sum_{k>=0} (-1)^k/binomial(2*k,k).

Equals Sum_{k>=0} (-1)^k*(k!)^2/(2*k)!.

cp's OEIS Frontend

A145433 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*n/binomial(2n,n).

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A157701 Decimal expansion of 2(14sigma+5)/625 where sigma = sqrt(5)*log(golden ratio).

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A307086 Decimal expansion of 4(5 - sqrt(5)log(phi))/25, where phi is the golden ratio (A001622).

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cp's OEIS Frontend

A145433 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*n/binomial(2n,n).

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A157701 Decimal expansion of 2*(14*sigma+5)/625 where sigma = sqrt(5)*log(golden ratio).

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A307086 Decimal expansion of 4*(5 - sqrt(5)*log(phi))/25, where phi is the golden ratio (A001622).

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A157701 Decimal expansion of 2(14sigma+5)/625 where sigma = sqrt(5)*log(golden ratio).

A307086 Decimal expansion of 4(5 - sqrt(5)log(phi))/25, where phi is the golden ratio (A001622).