A111877 -id:A111877 - cp's OEIS frontend

A111878 a(n) = A111877(n+1)/5.

1, 7, 21, 231, 3003, 3003, 51051, 969969, 969969, 22309287, 111546435, 334639305, 9704539845, 300840735195, 300840735195, 300840735195, 11131107202215, 11131107202215, 456375395290815, 19624141997505045, 19624141997505045

Offset: 0

Paul Barry, Aug 19 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A025547, A111877.

H:=HarmonicNumber; [Denominator((2*H(2*n+6) - H(n+3)))/15: n in [0..40]]; // G. C. Greubel, Jul 24 2023

With[{H=HarmonicNumber}, Table[Denominator[2*H[2*n+6] -H[n+3]]/15, {n, 0, 40}]] (* G. C. Greubel, Jul 24 2023 *)

h=harmonic_number; [denominator(2*h(2*n+6,1) - h(n+3,1))/15 for n in range(41)] # G. C. Greubel, Jul 24 2023

a(n) = (1/15)*denominator(digamma(n+7/2)/2 + log(2) + euler_gamma/2).
a(n) = denominator(f(n+2)/15), where f(n) = Sum_{j=0..n} 1/(2*j+1).
a(n) = (1/15) * denominator of ( 2*H_{2*n+6} - H_{n+3} ), where H_{n} is the n-th Harmonic number. - G. C. Greubel, Jul 24 2023

A350669 Numerators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

Original entry on oeis.org

1, 4, 23, 176, 563, 6508, 88069, 91072, 1593269, 31037876, 31730711, 744355888, 3788707301, 11552032628, 340028535787, 10686452707072, 10823198495797, 10952130239452, 409741429887649, 414022624965424, 17141894231615609, 743947082888833412, 750488463554668427, 35567319917031991744, 250947670863258378883, 252846595191840484708, 13497714685925233086599

Offset: 0

Wolfdieter Lang, Mar 16 2022

For the denominators see A350670.
This sequence coincides with A025550(n+1), for n = 0, 1, ..., 37. See the comments there.
Thanks to Ralf Steiner for sending me a paper where this and similar sums appear.

Hugo Pfoertner, Table of n, a(n) for n = 0..100
Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions. p.258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. p. 258.
Comparison to A025550 using Plot 2.

Cf. A001620, A025547, A025550, A111877 (denominators), A350670.

[Numerator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1)))/2: n in [0..40]]; // G. C. Greubel, Jul 24 2023

With[{H=HarmonicNumber}, Table[Numerator[2*H[2*n+2] -H[n+1]]/2 , {n,0,50}]] (* G. C. Greubel, Jul 24 2023 *)

a(n) = numerator(sum(j=0, n, 1/(2*j+1))); \\ Michel Marcus, Mar 16 2022

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

a(n) = numerator((Psi(n+3/2) + gamma + 2*log(2))/2), with the Digamma function Psi(z), and the Euler-Mascheroni constant gamma = A001620. See Abramowitz-Stegun, p. 258. 6.3.4.

a(n) = (1/2) * numerator of ( 2*H_{2*n+2} - H_{n+1} ), where H_{n} is the n-th Harmonic number. - G. C. Greubel, Jul 24 2023

cp's OEIS Frontend

A111878 a(n) = A111877(n+1)/5.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula