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

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

0,2

Comments

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.

References

Bruce C. Berndt, Ramanujan's Notebooks Part I, Springer-Verlag, 1985. See p. 108, Entry 15.

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 A025550 using Plot 2.

Formula

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

From Peter Bala, Nov 30 2025: (Start)

Let G(n) = Sum_{j = 0..floor((n-1)/2)} 1/(2*j+1) so that G(2*n+1) = A350669(n)/A350670(n). Then the e.g.f. Sum_{k >= 1} G(k)*x^k/k! = exp(x) * Sum_{k >= 1} (-2)^(k-1)*x^k/(k*k!). See Berndt, p. 108 Entry 15.

a(n) = numerator( Sum_{k = 1..2*n+1} binomial(2*n+1,k)*(-2)^(k-1)/k ). (End)

a(n)/A350670(n) ~ log(n)/2 + c, where c = gamma/2 + log(2) (A114864). - Amiram Eldar, Jul 06 2026

Maple

seq( numer(add( 1/(2*k + 1) , k = 0..n)), n = 0..25); # Peter Bala, Nov 30 2025

Mathematica

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

Prog

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

Crossrefs

Cf. A001620, A025547, A025550, A111877 (denominators), A114864, A350670.

Sequence in context: A083355 A141763 A025550 * A067545 A004041 A220353

Adjacent sequences: A350666 A350667 A350668 * A350670 A350671 A350672

Keyword

nonn,frac,easy,changed

Author

Wolfdieter Lang, Mar 16 2022

Status

approved