A101028 -id:A101028 - cp's OEIS frontend

A082687 Numerator of Sum_{k=1..n} 1/(n+k).

1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 7751493599, 225175759291, 13981692518567, 14000078506967, 98115155543129, 3634060848592973, 3637485804655193

Offset: 1

Benoit Cloitre, Apr 12 2003

Numerator of Sum_{k=0..n-1} 1/((k+1)(2k+1)) (denominator is A111876). - Paul Barry, Aug 19 2005
Numerator of the sum of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n). - Alexander Adamchuk, Apr 11 2006
Numerator of the 2n-th alternating harmonic number H'(2n) = Sum ((-1)^(k+1)/k, k=1..2n). H'(2n) = H(2n) - H(n), where H(n) = Sum_{k=1..n} 1/k is the n-th Harmonic Number. - Alexander Adamchuk, Apr 11 2006
a(n) almost always equals A117731(n) = numerator(n*Sum_{k=1..n} 1/(n+k)) = numerator(Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1)) but differs for n = 14, 53, 98, 105, 111, 114, 119, 164. - Alexander Adamchuk, Jul 16 2006
Sum_{k=1..n} 1/(n+k) = n!^2 *Sum_{j=1..n} (-1)^(j+1) /((n+j)!(n-j)!j). - Leroy Quet, May 20 2007
Seems to be the denominator of the harmonic mean of the first n hexagonal numbers. - Colin Barker, Nov 19 2014
Numerator of 2*n*binomial(2*n,n)*Sum_{k = 0..n-1} (-1)^k* binomial(n-1,k)/(n+k+1)^2. Cf. A049281. - Peter Bala, Feb 21 2017
From Peter Bala, Feb 16 2022: (Start)
2*Sum_{k = 1..n} 1/(n+k) = 1 + 1/(1*2)*(n-1)/(n+1) - 1/(2*3)*(n-1)*(n-2)/((n+1)*(n+2)) + 1/(3*4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + - .... Cf. A101028.
2*Sum_{k = 1..n} 1/(n+k) = n - (1 + 1/2^2)*n*(n-1)/(n+1) + (1/2^2 + 1/3^2)*n*(n-1)*(n-2)/((n+1)*(n+2)) - (1/3^2 + 1/4^2)*n*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + (1/4^2 + 1/5^2)*n*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) - + .... Cf. A007406 and A120778.
These identities allow us to extend the definition of Sum_{k = 1..n} 1/(n+k) to non-integral values of n. (End)

			H'(2n) = H(2n) - H(n) = {1/2, 7/12, 37/60, 533/840, 1627/2520, 18107/27720, 237371/360360, 95549/144144, 1632341/2450448, 155685007/232792560, ...}, where H(n) = A001008/A002805.
n=2: HilbertMatrix(n,n)
   1  1/2
  1/2 1/3
so a(2) = Numerator(1 + 1/2 + 1/2 + 1/3) = Numerator(7/3) = 7.
The n X n Hilbert matrix begins:
   1   1/2  1/3  1/4  1/5  1/6  1/7  1/8  ...
  1/2  1/3  1/4  1/5  1/6  1/7  1/8  1/9  ...
  1/3  1/4  1/5  1/6  1/7  1/8  1/9  1/10 ...
  1/4  1/5  1/6  1/7  1/8  1/9  1/10 1/11 ...
  1/5  1/6  1/7  1/8  1/9  1/10 1/11 1/12 ...
  1/6  1/7  1/8  1/9  1/10 1/11 1/12 1/13 ...

T. D. Noe, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Bisection of A058313, A082688 (denominators).
Cf. A001008, A002805, A005249, A007406, A058312.
Cf. A086881, A098118, A101028, A117731, A120778.

[Numerator((HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // G. C. Greubel, Jul 24 2023

a := n -> numer(harmonic(2*n) - harmonic(n)):
seq(a(n), n=1..20); # Peter Luschny, Nov 02 2017

Numerator[Sum[1/k,{k,1,2*n}] - Sum[1/k,{k,1,n}]] (* Alexander Adamchuk, Apr 11 2006 *)
Table[Numerator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 20}] (* Alexander Adamchuk, Apr 11 2006 *)
Table[HarmonicNumber[2 n] - HarmonicNumber[n], {n, 20}] // Numerator (* Eric W. Weisstein, Dec 14 2017 *)

a(n) = numerator(sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017

[numerator(harmonic_number(2*n,1) - harmonic_number(n,1)) for n in range(1,41)] # G. C. Greubel, Jul 24 2023

Limit_{n -> oo} Sum_{k=1..n} 1/(n+k) = log(2).
Numerator of Psi(2*n+1) - Psi(n+1). - Vladeta Jovovic, Aug 24 2003
a(n) = numerator((Sum_{k=1..2*n} 1/k) - Sum_{k=1..n} 1/k). - Alexander Adamchuk, Apr 11 2006
a(n) = numerator(Sum_{j=1..n} (Sum_{i=1..n} 1/(i+j-1))). - Alexander Adamchuk, Apr 11 2006
The o.g.f for Sum_{k=1..n} 1/(n+k) is f(x) = (sqrt(x)*log((1+sqrt(x))/(1-sqrt(x))) + log(1-x))/(2*x*(1-x)).

A120778 Numerators of partial sums of Catalan numbers scaled by powers of 1/4.

Original entry on oeis.org

1, 5, 11, 93, 193, 793, 1619, 26333, 53381, 215955, 436109, 3518265, 7088533, 28539857, 57414019, 1846943453, 3711565741, 14911085359, 29941580393, 240416274739, 482473579583, 1936010885087, 3883457090629, 62306843256889

Offset: 0

Wolfdieter Lang, Jul 20 2006

For denominators see A120777.
From the expansion of 0 = sqrt(1-1) = 1 - (1/2)*Sum_{k>=0} C(k)/4^k one has r:=lim_{n->infinity} r(n) = 2, with the partial sums r(n) defined below.
The series a(n)/A046161(n+1) is absolutely convergent to 1. - Ralf Steiner, Feb 16 2017
If n >= 1 it appears a(n-1) is equal to the difference between the denominator and the numerator of the ratio (2n)!!/(2n-1)!!. In particular a(n-1) is the difference between the denominator and the numerator of the ratio A001147(2n-1)/A000165(2n). See examples. - Anthony Hernandez, Feb 05 2020
From Peter Bala, Feb 16 2022: (Start)
Sum_{k = 0..n-1} Catalan(k)/4^k = (1/4^n)*(2*n)*binomial(2*n,n) *( 1 - 1/(1*2)*(n-1)/(n+1) - 1/(2*3)*(n-1)*(n-2)/((n+1)*(n+2)) - 1/(3*4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) - ... ). Cf. A082687 and A101028.
This identity allows us to extend the definition of Sum_{k = 0..n} Catalan(k)/4^k to non-integral values of n. (End)

			Rationals r(n): [1, 5/4, 11/8, 93/64, 193/128, 793/512, 1619/1024, 26333/16384, ...].
From _Anthony Hernandez_, Feb 05 2020: (Start)
For n = 4. The 4th even number is 8, and 8!!/(8-1)!! = 128/35, so a(4-1) = a(3) = 128 - 35 = 93.
For n = 7. The 7th even number is 14, and 14!!/(14-1)!! = 2048/429, so a(7-1) = a(6) = 2048 - 429 = 1619. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Wolfdieter Lang, Rationals r(n) and limit 2

Factors of A160481. Cf. A120777 (denominators), A082687, A101028, A141244.

[Numerator(2*(1-Binomial(2*n+2,n+1)/4^(n+1))): n in [0..25]]; // Vincenzo Librandi, Feb 17 2017

a := n -> 2^(2*(n+1) - add(i, i=convert(n+1, base, 2)))* (1-((n+1/2)!)/(sqrt(Pi)*(n+1)!)): seq(simplify(a(n)), n=0..23); # Peter Luschny, Dec 21 2017

f[n_] := f[n] = Numerator[(4/Pi) (n + 1) Integrate[x^n*ArcSin[Sqrt[x]], {x, 0, 1}]]; Array[f, 23, 0] (* Robert G. Wilson v, Jan 03 2011 *)
a[n_] := 2^(2(n+1) - DigitCount[n+1,2,1])(1 - ((n+1/2)!)/(Sqrt[Pi](n+1)!));
Table[a[n], {n, 0, 23}] (* Peter Luschny, Dec 21 2017 *)

a(n) = numerator(r(n)), with the rationals r(n):=Sum_{k = 0..n} C(k)/4^k with C(k) := A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
r(n) = (4/Pi)*(n+1)*Integral_{x = 0..1} x^n*arcsin(sqrt(x)) dx. - Groux Roland, Jan 03 2011
r(n) = 2*(1 - binomial(2*n+2,n+1)/4^(n+1)). - Groux Roland, Jan 04 2011
a(n) = A141244(2n+2) = A141244(2n+3) (conjectural). - Greg Martin, Aug 16 2014, corrected by M. F. Hasler, Aug 18 2014
From Peter Luschny, Dec 21 2017: (Start)
a(n) = numerator(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)).
a(n) = 2^(2*(n+1) - HammingWeight(n+1))*(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)). (End)

A101627 Numerator of partial sums of a certain series.

Original entry on oeis.org

1, 39, 241, 34883, 14039, 1516871, 7601151, 875425657, 7887002813, 7095769757767, 14199583385459, 75087685321529, 75113436870869, 927229349730873529, 927436191807263569, 305182576081725442901, 23479178371879154033, 37713848011377144613, 37717984058802320713, 135759786815564675620247

Offset: 1

Wolfdieter Lang, Dec 23 2004

The denominators are given in A101628.
Second member (m=3) of a family defined in A101028.
The limit s=lim(s(n),n->infinity) with the s(n) defined below equals 8*sum(zeta(2*k+1)/3^(2*k),k=1..infinity) with Euler's (or Riemann's) zeta function. This limit is 12*(log(3)-1) = 1.18334746...; see the Abramowitz-Stegun (given in A101028) reference p. 259, eq. 6.3.15 with z=1/3 together with p. 258.

			s(3)= 24*(1/(2*3*4)+ 1/(5*6*7) + 1/(8*9*10)) = 241/210, hence a(3)=241 and A101628(3)=210.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, Rationals s(n) and more.

Cf. A101028 (m=2), A101629 (m=4), A101631 (m=5).

Cf. A101628 (denominators).

Numerator[Accumulate[Table[8/(9k^3-k),{k,20}]]]

a(n) = numerator(24*sum(k=1, n, 1/((3*k-1)*(3*k)*(3*k+1))));

a(n)=numerator(s(n)) with s(n)=24*sum(1/((3*k-1)*(3*k)*(3*k+1)), k=1..n).

More terms from Michel Marcus, Mar 01 2022

A101629 Numerator of partial sums of a certain series.

Original entry on oeis.org

1, 47, 6931, 238657, 4563655, 526760263, 45934377581, 2852342564497, 105651280880749, 4335127472172929, 186521117762900387, 61393482232562091673, 3255023127143379846869, 3255958701070954680689

Offset: 1

Wolfdieter Lang, Dec 23 2004

The denominators are given in A101630.
Third member (m=4) of a family defined in A101028.
The limit s=lim(s(n),n->infinity) with the s(n) defined below equals 15*sum(Zeta(2*k+1)/4^(2*k),k=1..infinity) with Euler's (or Riemann's) zeta function. This limit is 15*(3*log(2)-2) = 1.1916231251...; see the Abramowitz-Stegun reference (given in A101028) p. 259, eq. 6.3.15 with z=1/4 together with p. 258.

			s(3)= 60*(1/(3*4*5)+ 1/(7*8*9) + 1/(11*12*13)) = 6931/6006, hence
a(3)=6931 and A101630(3)=6006.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang: Rationals s(n) and more.

Cf. A101028, A101627, A101631, members 2, 3, 5, resp.

a(n)=numerator(s(n)) with s(n)=60*sum(1/((4*k-1)*(4*k)*(4*k+1)), k=1..n) = 15*sum(1/((4*k-1)*k*(4*k+1)), k=1..n).

A101631 Numerator of partial sums of a certain series.

Original entry on oeis.org

1, 37, 1069, 20575, 1346153, 1214756107, 20699705479, 850029466379, 19572345658457, 137116980686111, 411600123273343, 1482039573988769177, 456179332236626381, 32398234503565880731, 1199020509231104363863

Offset: 1

Wolfdieter Lang, Dec 23 2004

The denominators are given in A101632.
Third member (m=5) of a family defined in A101028.
The limit s = lim_{n->oo} s(n) with the s(n) defined below equals 24*Sum_{k>=1} zeta(2*k+1)/5^(2*k) with Euler's (or Riemann's) zeta function. This limit is -24*(gamma + Psi(1/5) + 5/2 + Pi*cot(Pi/5)/2) = 1.1954056019...; see a comment in A101028 following from the Abramowitz-Stegun reference (given in A101028) p. 259, eq. 6.3.15 with z=1/5 together with p. 258.

			s(3) = 120*(1/(4*5*6) + 1/(9*10*11) + 1/(14*15*16)) = 1069/924, hence a(3)=1069 and A101632(3)=924.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang: Rationals s(n,5) and more.

Cf. A101028, A101627, A101629, members 2, 3, 4, resp.

a(n) = numerator(s(n)) where s(n) = 120*Sum_{k=1..n} 1/((5*k-1)*(5*k)*(5*k+1)) = 24*Sum_{k=1..n} 1/((5*k-1)*k*(5*k+1)).

A101029 Denominator of partial sums of a certain series.

Original entry on oeis.org

1, 10, 70, 420, 4620, 60060, 60060, 408408, 7759752, 38798760, 892371480, 4461857400, 13385572200, 55454513400, 1719089915400, 3438179830800, 24067258815600, 890488576177200, 890488576177200, 36510031623265200, 1569931359800403600, 1569931359800403600, 73786773910618969200

Offset: 1

Wolfdieter Lang, Dec 17 2004

The numerators are given in A101028.

One third of the denominator of the finite differences of the series of sums of all matrix elements of n X n Hilbert matrix M(i,j)=1/(i+j-1) (i,j = 1..n). - Alexander Adamchuk, Apr 11 2006

			n=2: HilbertMatrix[n,n]
   1  1/2
  1/2 1/3
so a(1) = (1/3)*denominator((1 + 1/2 + 1/2 + 1/3) - 1) = (1/3)*denominator(4/3) = 1.
The n X n Hilbert matrix begins:
   1  1/2 1/3 1/4 1/5  1/6  1/7  1/8 ...
  1/2 1/3 1/4 1/5 1/6  1/7  1/8  1/9 ...
  1/3 1/4 1/5 1/6 1/7  1/8  1/9  1/10 ...
  1/4 1/5 1/6 1/7 1/8  1/9  1/10 1/11 ...
  1/5 1/6 1/7 1/8 1/9  1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...

Eric Weisstein's World of Mathematics, Hilbert Matrix.
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A098118, A086881, A005249, A001008, A002805.

Cf. A101028 (numerators).

Denominator[Table[Sum[1/(i + j - 1), {i, n}, {j, n}], {n,2, 27}]-Table[Sum[1/(i + j - 1), {i, n}, {j, n}], {n, 26}]]/3 (* Alexander Adamchuk, Apr 11 2006 *)

a(n) = denominator(3*sum(k=1, n, 1/((2*k-1)*k*(2*k+1)))); \\ Michel Marcus, Feb 28 2022

a(n) = denominator(s(n)) with s(n) = 3*Sum_{k=1..n} 1/((2*k-1)*k*(2*k+1)). See A101028 for more information.

a(n) = (1/3)*denominator((Sum_{i=1..n+1} Sum_{j=1..n+1} 1/(i+j-1)) - (Sum_{i=1..n} Sum_{j=1..n} 1/(i+j-1))). a(n) = (1/3)*denominator(H(2*n+1) + H(2*n) - 2*H(n)), where H(n) = Sum_{k=1..n} 1/k is a harmonic number, H(n) = A001008/A002805. - Alexander Adamchuk, Apr 11 2006

More terms from Michel Marcus, Feb 28 2022

cp's OEIS Frontend

A082687 Numerator of Sum_{k=1..n} 1/(n+k).

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A120778 Numerators of partial sums of Catalan numbers scaled by powers of 1/4.

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A101627 Numerator of partial sums of a certain series.

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A101629 Numerator of partial sums of a certain series.

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A101631 Numerator of partial sums of a certain series.

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A101029 Denominator of partial sums of a certain series.

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