cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A076175 Denominator of Sum_{i+j+k=n, i,j,k>=1} (i*j)/k.

Original entry on oeis.org

1, 1, 1, 2, 3, 12, 15, 30, 21, 168, 126, 1260, 990, 792, 1287, 18018, 5005, 80080, 68068, 58344, 50388, 201552, 176358, 3879876, 3432198, 27457584, 24515700, 637408200, 573667380, 16062686640, 29113619535, 26466926850, 24131609775
Offset: 1

Views

Author

Benoit Cloitre, Nov 01 2002

Keywords

Crossrefs

Cf. A076174.

Programs

  • PARI
    a(n)=denominator(sum(i=1,n,sum(j=1,n,sum(k=1,n,if(n-i-j-k,0,1)*i*j/k))))

A354894 a(n) is the numerator of the n-th hyperharmonic number of order n.

Original entry on oeis.org

1, 5, 47, 319, 1879, 20417, 263111, 261395, 8842385, 33464927, 166770367, 3825136961, 19081066231, 57128792093, 236266661971, 7313175618421, 14606816124167, 102126365345729, 3774664307989373, 3771059091081773, 154479849447926113, 6637417807457499259, 6632660439700528339
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 10 2022

Keywords

Examples

			1, 5/2, 47/6, 319/12, 1879/20, 20417/60, 263111/210, 261395/56, 8842385/504, ...

References

  • J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 258.

Links

Crossrefs

Cf. A001008, A001700, A027612, A058806, A076174, A124837, A354895 (denominators).
Differs from A049281.

Programs

  • Mathematica
    Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 23}] // Numerator
Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 23}] // Numerator
  • PARI
    H(n) = sum(i=1, n, 1/i);
a(n) = numerator(binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1))); \\ Michel Marcus, Jun 10 2022
  • Python
    from math import comb
from sympy import harmonic
def A354894(n): return (comb(2*n-1,n-1)*(harmonic(2*n-1)-harmonic(n-1))).p # Chai Wah Wu, Jun 18 2022

Formula

a(n) is the numerator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.
a(n) is the numerator of binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1)), where H(n) is the n-th harmonic number.
a(n) / A354895(n) ~ log(2) * 2^(2*n-1) / sqrt(Pi * n).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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