A099946 -id:A099946 - cp's OEIS frontend

A363154 Triangle read by rows. The Hadamard product of A173018 and A349203.

1, 1, 0, 2, 1, 0, 3, 4, 1, 0, 12, 33, 22, 3, 0, 10, 52, 66, 26, 2, 0, 60, 570, 1208, 906, 228, 10, 0, 105, 1800, 5955, 7248, 3573, 600, 15, 0, 280, 8645, 42930, 78095, 62476, 21465, 2470, 35, 0, 252, 14056, 102256, 264702, 312380, 176468, 43824, 3514, 28, 0

Offset: 0

Peter Luschny, May 21 2023

			Triangle T(n, k) starts:
[0]   1;
[1]   1,    0;
[2]   2,    1,     0;
[3]   3,    4,     1,     0;
[4]  12,   33,    22,     3,     0;
[5]  10,   52,    66,    26,     2,     0;
[6]  60,  570,  1208,   906,   228,    10,    0;
[7] 105, 1800,  5955,  7248,  3573,   600,   15,  0;
[8] 280, 8645, 42930, 78095, 62476, 21465, 2470, 35, 0;

Peter Luschny, A combinatorial triangle for the Bernoulli numbers, MathOverflow 2023.

Cf. A173018, A349203, A002944 (column 0), A099946, A362994 (alternating row sums), A362990 (row sums).

A173018 := (n, k) -> combinat[eulerian1](n, k):
A349203 := (n, k) -> ilcm(seq(binomial(n, j), j = 0..n)) / binomial(n, k):
A363154 := (n, k) -> A173018(n, k) * A349203(n, k):
for n from 0 to 8 do seq(A363154(n, k), k = 0..n) od;

T(n, k) = A173018(n, k) * A349203(n, k).

Sum_{k=0..n} (-1)^k * T(n, k) = lcm(1, 2, ..., n+1)*Bernoulli(n, 1) = A362994(n).

A355947 a(n) = Sum_{k=1..n} (n+1-k)*floor(n/k).

Original entry on oeis.org

0, 1, 5, 12, 25, 39, 65, 87, 124, 161, 210, 249, 328, 377, 450, 531, 630, 698, 825, 903, 1047, 1169, 1295, 1393, 1609, 1740, 1893, 2056, 2269, 2400, 2679, 2822, 3070, 3277, 3486, 3709, 4082, 4260, 4498, 4748, 5136, 5336, 5744, 5956, 6312, 6686, 6984, 7218, 7772

Offset: 0

Tamas Sandor Nagy, Jul 21 2022

In the sum, the divisors are k = 1..n and the multipliers are the reverse n+1-k = n .. 1.
From Thomas Scheuerle, Jul 21 2022: (Start)
Without the floor operation a(n) would be A027457(n+1)/A099946(n+1) = (H(n+1) - 1)*(n^2+n), where H(n) is the n-th harmonic number.
What is lim_{n->oo} ((A027457(n+1)/A099946(n+1) - a(n))/(n^2+n))? It appears to be close to 0.2452... . (End)
This limit is equal to Pi^2/12 - gamma = 0.24525136852258... - Vaclav Kotesovec, Jul 23 2022

			For n=5, the sum is formed:
  k = 1..n:                1   2   3   4   5
  floor(n/k):              5   2   1   1   1
  n+1-k = n..1:            5   4   3   2   1
  floor(n/k)*(n+1-k):     25   8   3   2   1
                          __________________
  a(5) =                  25 + 8 + 3 + 2 + 1 = 39

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A006218, A024916, A027457, A099946, A143274.

a[n_] := Sum[(n+1-k) * Floor[n/k], {k, 1, n}]; Array[a, 50, 0] (* Amiram Eldar, Jul 22 2022 *)

a(n) = sum(k=1, n, (n+1-k)*floor(n/k)) \\ Rémy Sigrist, Jul 21 2022

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k*(k-(k-1)*x-x^k)/(1-x^k)^2)/(1-x)^2)) \\ Seiichi Manyama, Jul 24 2022

from math import isqrt
def A355947(n): return (s:=isqrt(n))**2*(s-(n<<1)-1)+sum((q:=n//k)*((n<<2)-(k<<1)-q+3) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 24 2023

From Vaclav Kotesovec, Jul 23 2022: (Start)
For n > 0, a(n) = (n+1)*A006218(n) - A024916(n).
a(n) ~ n*(n+1)*(log(n) + 2*gamma - 1) - n^2*Pi^2/12, where gamma is the Euler-Mascheroni constant A001620. (End)
(1/(1-x)^2) * Sum_{k>0} x^k * (k - (k-1)*x - x^k)/(1-x^k)^2. - Seiichi Manyama, Jul 24 2022

More terms from Rémy Sigrist, Jul 21 2022

A119936 Least common multiple (LCM) of denominators of the rows of the triangle of rationals A119935/A119932.

Original entry on oeis.org

1, 8, 108, 576, 18000, 21600, 1234800, 5644800, 57153600, 63504000, 8452382400, 9220780800, 1688171284800, 1818030614400, 1947889944000, 8310997094400, 2551995545299200, 2702112930316800, 1029655143835718400

Offset: 1

Wolfdieter Lang, Jul 20 2006

The triangle of rationals is the matrix cube of the matrix with elements a(i,j) = 1/i if j <= i, 0 if j > i.

Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012.

Distinct from A246498.

A027447(i,j)= a(i)* A119935(i,j)/A119932(i,j) .
a(n) = lcm_{m=1..n} seq(A119932(n,m)), n >= 1.
a(n)/n^3 = A027451(n) = A002944(n)^2 (the second equation is a conjecture).
a(n)/n^3 = (A099946(n)*(n-1))^2, n >= 2 (from the conjecture).

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A363154 Triangle read by rows. The Hadamard product of A173018 and A349203.

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A355947 a(n) = Sum_{k=1..n} (n+1-k)*floor(n/k).

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A119936 Least common multiple (LCM) of denominators of the rows of the triangle of rationals A119935/A119932.

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