A124838 -id:A124838 - cp's OEIS frontend

A213999 Denominators of the triangle of fractions read by rows: pf(n,0) = 1, pf(n,n) = 1/(n+1) and pf(n+1,k) = pf(n,k) + pf(n,k-1) with 0 < k < n; denominators: A213998.

1, 1, 2, 1, 2, 3, 1, 2, 6, 4, 1, 2, 3, 12, 5, 1, 2, 6, 12, 60, 6, 1, 2, 3, 4, 10, 20, 7, 1, 2, 6, 12, 20, 20, 140, 8, 1, 2, 3, 12, 15, 10, 35, 280, 9, 1, 2, 6, 4, 20, 30, 70, 280, 2520, 10, 1, 2, 3, 12, 10, 12, 21, 56, 252, 2520, 11, 1, 2, 6, 12, 60, 60, 84, 168, 504, 2520, 27720, 12

Offset: 0

Reinhard Zumkeller, Jul 03 2012

T(n,0) = 1;
T(n,1) = A007395(n) for n > 0;
T(n,2) = A010704(n) for n > 1;
T(n,n-3) = A124838(n-2) for n > 2;
T(n,n-2) = A027611(n-1) for n > 1;
T(n,n-1) = A002805(n) for n > 0;
T(n,n) = n + 1;
A003418(n+1) = least common multiple of n-th row;
A214075(n,k) = floor(A213998(n,k) / T(n,k)).

			See A213998.

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

import Data.Ratio ((%), denominator, Ratio)
a213999 n k = a213999_tabl !! n !! k
a213999_row n = a213999_tabl !! n
a213999_tabl = map (map denominator) $ iterate pf [1] where
   pf row = zipWith (+) ([0] ++ row) (row ++ [-1 % (x * (x + 1))])
            where x = denominator $ last row

T[, 0] = 1; T[n, n_] := 1/(n + 1);
T[n_, k_] := T[n, k] = T[n - 1, k] + T[n - 1, k - 1];
Table[T[n, k] // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 10 2021 *)

A124837 Numerators of third-order harmonic numbers (defined by Conway and Guy, 1996).

Original entry on oeis.org

1, 7, 47, 57, 459, 341, 3349, 3601, 42131, 44441, 605453, 631193, 655217, 1355479, 23763863, 24444543, 476698557, 162779395, 166474515, 34000335, 265842403, 812400067, 20666950267, 21010170067, 192066102203, 194934439103

Offset: 1

Jonathan Vos Post, Nov 10 2006

Denominators are A124838. All fractions reduced. Thanks to Jonathan Sondow for verifying these calculations. He suggests that the equivalent definition in terms of first order harmonic numbers may be computationally simpler. We are happy with the description of A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n, but baffled by the description of A027611.
From Alexander Adamchuk, Nov 11 2006: (Start)
a(n) is the numerator of H(n, (3)) = Sum_{m=1..n} Sum_{k=1..m} HarmonicNumber(k).
Denominators are listed in A124838.
p divides a(p-5) for prime p > 5.
Primes are listed in A129880.
Numbers k such that a(k) is prime are listed in A129881. (End)

			a(1) = 1 = numerator of 1/1.
a(2) = 7 = numerator of 1/1 + 5/2 = 7/2.
a(3) = 47 = numerator of 7/2 + 13/3 = 47/6.
a(4) = 57 = numerator of 47/6 + 77/12 = 57/4.
a(5) = 549 = numerator of 57/4 + 87/10 = 549/20.
a(6) = 341 = numerator of 549/20 + 223/20 = 341/10
a(7) = 3349 = numerator of 341/10 + 481/35 = 3349/70.
a(8) = 88327 = numerator of 3349/70 + 4609/280 = 88327/1260.
a(9) = 3844 = numerator of 88327/1260 + 4861/252 = 3844/45.
a(10) = 54251 = numerator of 3844/45 + 55991/2520 = 54251/504, or, untelescoping:
a(10) = 54251 = numerator of 1/1 + 5/2 + 13/3 + 77/12 + 87/10 + 223/20 + 481/35 + 4609/252 + 4861/252 + 55991/2520 = 54251/504.

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Harmonic Number. See equation for third order harmonic numbers.

Cf. A027611, A027612, A124838.

Cf. A001008, A002805, A067657, A056903, A124878, A124879, A124837, A129880, A129881.

a124837 n = a213998 (n + 2) (n - 1) -- Reinhard Zumkeller, Jul 03 2012

Table[Numerator[(n+2)!/2!/n!*Sum[1/k,{k,3,n+2}]],{n,1,30}] (* Alexander Adamchuk, Nov 11 2006 *)

A124837(n)/A124838(n) = Sum{i=1..n} A027612(n)/A027611(n+1).
From Alexander Adamchuk, Nov 11 2006: (Start)
a(n) = numerator(Sum_{m=1..n} Sum_{l=1..m} Sum_{k=1..l} 1/k).
a(n) = numerator(((n+2)!/(2!*n!)) * Sum_{k=3..n+2} 1/k).
a(n) = numerator(((n+2)*(n+1)/2) * Sum_{k=3..n+2} 1/k). (End)
a(n) = numerator(Sum_{k=0..n-1} (-1)^k*binomial(-3,k)/(n-k)). - Gary Detlefs, Jul 18 2011
a(n) = A213998(n+2,n-1). - Reinhard Zumkeller, Jul 03 2012

Corrected and extended by Alexander Adamchuk, Nov 11 2006

A354895 a(n) is the denominator of the n-th hyperharmonic number of order n.

Original entry on oeis.org

1, 2, 6, 12, 20, 60, 210, 56, 504, 504, 660, 3960, 5148, 4004, 4290, 34320, 17680, 31824, 302328, 77520, 813960, 8953560, 2288132, 27457584, 49031400, 12498200, 168725700, 42948360, 10925460, 163881900, 2540169450, 645122400, 327523680, 5567902560, 1412149200

Offset: 1

Ilya Gutkovskiy, Jun 10 2022

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

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

Robert Israel, Table of n, a(n) for n = 1..3764
Eric Weisstein's World of Mathematics, Harmonic Number
Wikipedia, Hyperharmonic number

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

N:= 100: # for a(1)..a(N)
H:= ListTools:-PartialSums([seq(1/i,i=1..2*N-1)]):
f:= n -> denom(binomial(2*n-1,n-1)*(H[2*n-1]-H[n-1])):
f(1):= 1:
map(f, [$1..N]); # Robert Israel, Jul 10 2023

Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 35}] // Denominator
Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 35}] // Denominator

H(n) = sum(i=1, n, 1/i);
a(n) = denominator(binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1))); \\ Michel Marcus, Jun 10 2022

from math import comb
from sympy import harmonic
def A354895(n): return (comb(2*n-1,n-1)*(harmonic(2*n-1)-harmonic(n-1))).q # Chai Wah Wu, Jun 18 2022

a(n) is the denominator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.
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.
A354894(n) / a(n) ~ log(2) * 2^(2*n-1) / sqrt(Pi * n).

A316297 a(n) = n! times the denominator of the n-th harmonic number H(n).

Original entry on oeis.org

1, 4, 36, 288, 7200, 14400, 705600, 11289600, 914457600, 9144576000, 1106493696000, 13277924352000, 2243969215488000, 31415569016832000, 471233535252480000, 15079473128079360000, 4357967734014935040000, 26147806404089610240000, 9439358111876349296640000

Offset: 1

Matthew Campbell, Jun 29 2018

nonn

			a(4) = 4! * A002805(4) = 24 * 12 = 288.

Cf. A000142, A001008, A002805, A027611, A027612, A124837, A124838.

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> denom(H(n))*n!:
seq(a(n), n=1..20);  # Alois P. Heinz, Jul 21 2018

a[n_] := n! Denominator@HarmonicNumber@n; Array[a, 18] (* Robert G. Wilson v, Jun 30 2018 *)

a(n) = n! * denominator(sum(k=1, n, 1/k)); \\ Michel Marcus, Aug 12 2018

a(n) = A000142(n) * A002805(n).

cp's OEIS Frontend

A213999 Denominators of the triangle of fractions read by rows: pf(n,0) = 1, pf(n,n) = 1/(n+1) and pf(n+1,k) = pf(n,k) + pf(n,k-1) with 0 < k < n; denominators: A213998.

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A124837 Numerators of third-order harmonic numbers (defined by Conway and Guy, 1996).

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A354895 a(n) is the denominator of the n-th hyperharmonic number of order n.

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A316297 a(n) = n! times the denominator of the n-th harmonic number H(n).

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Maple

Mathematica

PARI

Formula