A350670 Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725, 96845140757687397075, 96845140757687397075, 5132792460157432044975

Offset

0,2

Comments

For the numerators see A350669.

This sequence coincides with A025547(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.

Links

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 A025547 using Plot 2.

Formula

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

Mathematica

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

Prog

(PARI) a(n) = denominator(sum(j=0, n, 1/(2*j+1))); \\ Michel Marcus, Mar 16 2022
(Magma) [Denominator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1))): n in [0..40]]; // G. C. Greubel, Jul 24 2023
(SageMath) [denominator(2*harmonic_number(2*n+2, 1) - harmonic_number(n+1, 1)) for n in range(41)] # G. C. Greubel, Jul 24 2023

Crossrefs

Cf. A001620, A025547, A025550, A350669 (numerators).

Sequence in context: A145624 A025547 A352395 * A220747 A088989 A001801

Adjacent sequences: A350667 A350668 A350669 * A350671 A350672 A350673

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Mar 16 2022

Status

approved