A354895 - OEIS

%I #19 Jul 11 2023 08:59:31

%S 1,2,6,12,20,60,210,56,504,504,660,3960,5148,4004,4290,34320,17680,

%T 31824,302328,77520,813960,8953560,2288132,27457584,49031400,12498200,

%U 168725700,42948360,10925460,163881900,2540169450,645122400,327523680,5567902560,1412149200

%N a(n) is the denominator of the n-th hyperharmonic number of order n.

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

%H Robert Israel, <a href="/A354895/b354895.txt">Table of n, a(n) for n = 1..3764</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyperharmonic_number">Hyperharmonic number</a>

%F a(n) is the denominator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.

%F a(n) is the denominator 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(n) / a(n) ~ log(2) * 2^(2*n-1) / sqrt(Pi * n).

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

%p N:= 100: # for a(1)..a(N)

%p H:= ListTools:-PartialSums([seq(1/i,i=1..2*N-1)]):

%p f:= n -> denom(binomial(2*n-1,n-1)*(H[2*n-1]-H[n-1])):

%p f(1):= 1:

%p map(f, [$1..N]); # _Robert Israel_, Jul 10 2023

%t Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 35}] // Denominator

%t Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 35}] // Denominator

%o (PARI) H(n) = sum(i=1, n, 1/i);

%o a(n) = denominator(binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1))); \\ _Michel Marcus_, Jun 10 2022

%o (Python)

%o from math import comb

%o from sympy import harmonic

%o def A354895(n): return (comb(2*n-1,n-1)*(harmonic(2*n-1)-harmonic(n-1))).q # _Chai Wah Wu_, Jun 18 2022

%Y Cf. A001700, A002805, A027611, A058806, A076175, A124838, A354894 (numerators).

%K nonn,frac

%O 1,2

%A _Ilya Gutkovskiy_, Jun 10 2022