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

%I A354894 #14 Feb 16 2025 08:34:03
%S A354894 1,5,47,319,1879,20417,263111,261395,8842385,33464927,166770367,
%T A354894 3825136961,19081066231,57128792093,236266661971,7313175618421,
%U A354894 14606816124167,102126365345729,3774664307989373,3771059091081773,154479849447926113,6637417807457499259,6632660439700528339
%N A354894 a(n) is the numerator of the n-th hyperharmonic number of order n.
%D A354894 J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 258.
%H A354894 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%H A354894 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyperharmonic_number">Hyperharmonic number</a>
%F A354894 a(n) is the numerator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.
%F A354894 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.
%F A354894 a(n) / A354895(n) ~ log(2) * 2^(2*n-1) / sqrt(Pi * n).
%e A354894 1, 5/2, 47/6, 319/12, 1879/20, 20417/60, 263111/210, 261395/56, 8842385/504, ...
%t A354894 Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 23}] // Numerator
%t A354894 Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 23}] // Numerator
%o A354894 (PARI) H(n) = sum(i=1, n, 1/i);
%o A354894 a(n) = numerator(binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1))); \\ _Michel Marcus_, Jun 10 2022
%o A354894 (Python)
%o A354894 from math import comb
%o A354894 from sympy import harmonic
%o A354894 def A354894(n): return (comb(2*n-1,n-1)*(harmonic(2*n-1)-harmonic(n-1))).p # _Chai Wah Wu_, Jun 18 2022
%Y A354894 Cf. A001008, A001700, A027612, A058806, A076174, A124837, A354895 (denominators).
%Y A354894 Differs from A049281.
%K A354894 nonn,frac
%O A354894 1,2
%A A354894 _Ilya Gutkovskiy_, Jun 10 2022