A110625
Numerator of b(n) = -Sum_{k=1..n} A037861(k)/((2*k)*(2*k+1)), where A037861(k) = (number of 0's) - (number of 1's) in the binary representation of k.
Original entry on oeis.org
1, 1, 3, 101, 5807, 77801, 82949, 170636, 170636, 170636, 363113, 363113, 84848, 710567, 22435781, 3901243741, 27210449083, 1003538672911, 248595095590537, 10165684261926701, 438167567023512863, 439119040574907047
Offset: 1
a(3) = 3 because b(3) = 1/6 + 0 + 1/21 = 3/14.
The first few fractions b(n) are 1/6, 1/6, 3/14, 101/504, 5807/27720, 77801/360360, 82949/360360, ... = A110625/A110626. - _Petros Hadjicostas_, May 15 2020
- Petros Hadjicostas, Table of n, a(n) for n = 1..120
- Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004.
- Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005), 61-65.
- Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005.
- Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
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a(n) = numerator(-sum(k=1, n, (#binary(k) - 2*hammingweight(k))/(2*k*(2*k+1)))); \\ Petros Hadjicostas, May 15 2020
A110626
Denominator of b(n) = -Sum_{k=1..n} A037861(k)/((2*k)*(2*k+1)), where A037861(k) = (number of 0's) - (number of 1's) in the binary representation of k.
Original entry on oeis.org
6, 6, 14, 504, 27720, 360360, 360360, 765765, 765765, 765765, 1601145, 1601145, 369495, 3061530, 94907430, 16703707680, 116925953760, 4326260289120, 1068586291412640, 43812037947918240, 1883917631760484320
Offset: 1
a(3) = 14 because b(3) = 1/6 + 0 + 1/21 = 3/14.
The first few fractions b(n) are 1/6, 1/6, 3/14, 101/504, 5807/27720, 77801/360360, 82949/360360, ... = A110625/A110626. - _Petros Hadjicostas_, May 15 2020
- Petros Hadjicostas, Table of n, a(n) for n = 1..100
- Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004.
- Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005), 61-65.
- Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005.
- Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
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a(n) = denominator(-sum(k=1, n, (#binary(k) - 2*hammingweight(k))/(2*k*(2*k+1))));\\ Petros Hadjicostas, May 15 2020
A073100
Denominator of b(n) = n * Sum_{k=2^n..2^(n+1)-1} (-1)^k/k.
Original entry on oeis.org
6, 210, 120120, 18050444111700, 118226688410282226751136160, 1112813007583117631616979100370019711878239390670756000, 1191035057635417333689929196555456096447880322064975132139675263681349241137859495385119040334214863238544000
Offset: 1
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a[n_] := Denominator[n * Sum[(-1)^k/k, {k, 2^n, 2^(n+1)-1}]]; Array[a, 7] (* Amiram Eldar, May 19 2022 *)
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a(n)=denominator( n*sum(k=2^n,2^(n+1)-1,(-1)^k/k))
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