A084623 -id:A084623 - cp's OEIS frontend

A339393 Denominators of the probability that when a stick is broken up at n-1 points independently and uniformly chosen at random along its length there exist 3 of the n pieces that can form a triangle.

1, 1, 4, 7, 28, 56, 88, 594, 5808, 415272, 8758464, 274431872, 12856077696, 905435186304, 481691519113728, 77763074616922464, 3824113551749834112, 1437016892446437662976, 165559472503434318118656, 146602912901791088694069888, 200050146291129782743679367168

Offset: 1

Amiram Eldar, Dec 04 2020

See A339392 for details.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000045, A001791, A084623, A234951, A243398, A339392 (numerators).

f = Table[k/(Fibonacci[k + 2] - 1), {k, 2, 20}]; Denominator[1 - FoldList[Times, 1, f]]

a(n) = denominator(1 - Product_{k=2..n} k/(Fibonacci(k+2)-1)).

A363399 Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (j + 1)^n), (tangent case).

Original entry on oeis.org

1, 3, 2, 7, 16, 9, 15, 88, 135, 64, 31, 416, 1296, 1536, 625, 63, 1824, 10206, 22528, 21875, 7776, 127, 7680, 72171, 262144, 453125, 373248, 117649, 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152, 511, 128512, 3057426, 25034752, 100000000, 218350080, 265180846, 167772160, 43046721

Offset: 0

Peter Luschny, May 31 2023

Here we give an inclusion-exclusion representation of 2^n*Euler(n, 1) = A155585(n), in A363398 we give such a representation for 2^n*Euler(n), and in A363400 one for the combined sequences.

			The triangle T(n, k) begins:
  [0]   1;
  [1]   3,     2;
  [2]   7,    16,      9;
  [3]  15,    88,    135,      64;
  [4]  31,   416,   1296,    1536,     625;
  [5]  63,  1824,  10206,   22528,   21875,     7776;
  [6] 127,  7680,  72171,  262144,  453125,   373248,  117649;
  [7] 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152;

Index entries for sequences related to Euler numbers.

Cf. A155585 (alternating row sums), A363397 (row sums), A126646 (column 0), A000169 (main diagonal), A163395 (central terms), A084623.

Cf. A363398 (secant case), A363400 (combined case).

P := (n, x) -> add(add(x^j*binomial(k, j)*(j + 1)^n, j=0..k)*2^(n - k), k = 0..n):
T := (n, k) -> coeff(P(n, x), x, k): seq(seq(T(n, k), k = 0..n), n = 0..8);

(* From  Detlef Meya, Oct 04 2023: (Start) *)
T[n_, k_] := (k+1)^n*(2^(n+1)-Sum[Binomial[n+1, j], {j, 0, k}]);
(* Or *)
T[n_, k_] := (k+1)^n*Binomial[n+1, k+1]*Hypergeometric2F1[1, k-n, k+2, -1];
Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* End *)

Sum_{k=0..n} (-1)^k * T(n, k) = 2^n*Euler(n, 1) = (-2)^n*Euler(n, 0) = A155585(n).
From  Detlef Meya, Oct 04 2023: (Start)
T(n, k) = (k + 1)^n*binomial(n + 1, k + 1)*hypergeom([1, k - n], [k + 2], -1).
T(n, k) = (k + 1)^n * (2^(n + 1) - add(binomial(n + 1, j), j=0..k)). (End)

A301631 Numerator of population variance of n-th row of Pascal's triangle.

Original entry on oeis.org

0, 0, 2, 1, 94, 122, 2372, 173, 50294, 56014, 983740, 266930, 18376812, 19624884, 333313544, 5500541, 5923399334, 6206260694, 103708093964, 27001710566, 1795265477444, 1860906681644, 30802090121144, 1988024895074, 524715115366844, 540193965134732, 8886200762228312

Offset: 0

N. J. A. Sloane and Chai Wah Wu, Mar 24 2018

Denominator of population variance of n-th row of Pascal's triangle is A191871(n+1) = A000265(n+1)^2.

			The first few population variances are 0, 0, 2/9, 1, 94/25, 122/9, 2372/49, 173, 50294/81, 56014/25, 983740/121, 266930/9, 18376812/169, 19624884/49, 333313544/225, 5500541, 5923399334/289, ...

Chai Wah Wu, Table of n, a(n) for n = 0..1659
Simon Demers, Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients, American Statistician, online: 19 Jan 2018.

Cf. A000108, A075101, A084623, A000265, A191871 (denominators), A301278, A301279, A054650, A301280.

a(n) = numerator(binomial(2*n,n)/(n+1) - 4^n/(n+1)^2); \\ Altug Alkan, Mar 25 2018

from fractions import Fraction
from sympy import binomial
def A301631(n):
    return (Fraction(int(binomial(2*n,n)))/(n+1) - Fraction(4**n)/(n+1)**2).numerator

a(n) = numerator of binomial(2n,n)/(n+1) - 4^n/(n+1)^2.

a(n) = A000108(n)*A000265(n+1)^2 - A075101(n+1)^2/4.

A320086 Triangle read by rows, 0 <= k <= n: T(n,k) is the denominator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; numerator is A320085.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 1, 2, 16, 16, 8, 8, 16, 16, 16, 4, 16, 1, 16, 4, 16, 64, 64, 64, 64, 64, 64, 64, 64, 16, 8, 8, 8, 1, 8, 8, 8, 16, 256, 256, 64, 64, 128, 128, 64, 64, 256, 256, 256, 32, 256, 16, 128, 1, 128, 16, 256, 32, 256

Offset: 0

Franck Maminirina Ramaharo, Oct 05 2018

			Triangle begins:
    1;
    1,   1;
    1,   1,   1;
    4,   4,   4,  4;
    2,   1,   1,  1,   2;
   16,  16,   8,  8,  16,  16;
   16,   4,  16,  1,  16,   4,  16;
   64,  64,  64, 64,  64,  64,  64, 64;
   16,   8,   8,  8,   1,   8,   8,  8,  16;
  256, 256,  64, 64, 128, 128,  64, 64, 256, 256;
  256,  32, 256, 16, 128,   1, 128, 16, 256,  32, 256;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Rita T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
Ron Goldman, Pyramid Algorithms. A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, Morgan Kaufmann Publishers, 2002, Chap. 5.
Eric Weisstein's World of Mathematics, Bernstein Polynomial
Wikipedia, Bernstein polynomial

Inspired by A141692.

Cf. A007318, A128433, A128434, A319861, A319862, A320085.

T:=proc(n,k) 2^(n-1)/gcd(n*(binomial(n-1,k-1)-binomial(n-1,k)),2^(n-1)); end proc: seq(seq(T(n,k),k=0..n),n=1..11); # Muniru A Asiru, Oct 06 2018

Table[Denominator[n*(Binomial[n-1, k-1] - Binomial[n-1, k])/2^(n-1)], {n, 0, 12}, {k, 0, n}]//Flatten

T(n, k) := 2^(n - 1)/gcd(n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 2^(n - 1))$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$

def A320086(n,k): return denominator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
flatten([[A320086(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021

T(n, k) = denominator of 2*A141692(n,k)/A000079(n).
T(n, k) = 2^(n-1)/gcd(n*(binomial(n-1, k-1) - binomial(n-1, k)), 2^(n-1)).
T(n, n-k) = T(n,k).
T(n, 0) = A084623(n), n > 0.
T(2*n+1, 1) = A000302(n).

A352485 Decimal expansion of the probability that when a unit interval is broken at two points uniformly and independently chosen at random along its length the lengths of the resulting three intervals are the altitudes of a triangle.

Original entry on oeis.org

2, 3, 2, 9, 8, 1, 4, 5, 8, 3, 1, 3, 6, 0, 9, 6, 9, 3, 3, 3, 4, 6, 3, 9, 7, 5, 9, 0, 8, 1, 4, 5, 3, 0, 2, 1, 0, 1, 8, 9, 6, 9, 6, 3, 8, 0, 9, 6, 6, 9, 5, 1, 7, 1, 4, 1, 6, 8, 1, 4, 6, 4, 9, 5, 8, 2, 1, 4, 6, 9, 1, 7, 1, 0, 6, 7, 1, 6, 7, 0, 7, 2, 6, 7, 5, 7, 6, 6, 3, 5, 2, 7, 3, 3, 2, 7, 8, 9, 2, 9, 7, 5, 1, 9, 3

Offset: 0

Amiram Eldar, Mar 18 2022

			0.23298145831360969333463975908145302101896963809669...

Mohammed Yaseen, Table of n, a(n) for n = 0..10000
Eugen J. Ionascu, Problem 11663, The American Mathematical Monthly, Vol. 119, No. 8 (2012), pp. 699-706; alternative link; Are Random Breaks the Altitudes of a Triangle?, Solution to Problem 116633, by David Farnsworth and James Marento, ibid., Vol. 121, No. 8 (2014), pp. 741-743.
Eugen J. Ionaşcu and Gabriel Prăjitură, Things to do with a broken stick, International Journal of Geometry, Vol. 2, No. 2 (2013), pp. 5-30; arXiv preprint, arXiv:1009.0890 [math.HO], 2010-2013.
Roberto Tauraso, Problem 11663.
Eric Weisstein's World of Mathematics, Altitude.

Cf. A001622, A002390.

Similar sequences: A016627, A019702, A084623, A243398, A339392, A339393, A352484.

RealDigits[24*Sqrt[5]*Log[GoldenRatio]/25 - 4/5, 10, 100][[1]]

Equals 12*sqrt(5)*log((3+sqrt(5))/2)/25 - 4/5.

Equals 24*sqrt(5)*log(phi)/25 - 4/5, where phi is the golden ratio (A001622).

A014964 a(n) = lcm(n, 2^(n-1)).

Original entry on oeis.org

1, 2, 12, 8, 80, 96, 448, 128, 2304, 2560, 11264, 6144, 53248, 57344, 245760, 32768, 1114112, 1179648, 4980736, 2621440, 22020096, 23068672, 96468992, 25165824, 419430400, 436207616, 1811939328

Offset: 1

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for sequences related to lcm's

Cf. A000027, A084623.

Table[LCM[n,2^(n-1)],{n,30}] (* Harvey P. Dale, Jun 03 2012 *)

a(n) = n * A084623(n). - Paul Curtz, May 21 2016.

A178623 Triangle T(n,m) read by rows: T(n,0)= prime(n); T(n,m)=1 if m>=1.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29

Offset: 0

Paul Curtz, May 31 2010

The sequence reflects a conjecture on the denominator of inverse Bernoulli polynomials in A178340: if the row index is one less than one of the primes in A008578, the row of denominators starts with that prime and contains 1's in the remaining entries.

[Row sums in A178252 are A159069(n+1), unless there is a common factor in numerator and denominator. The row sum over columns with index of the same parity as the row index in the table of fractions of the [x^m] B^{-1}(n,x) in A178252 are: 1, 1, 1/3+1=4/3, 1+1=2, 1/5+2+1=16/5, 1+10/3+1=16/3, 1/7+3+5+1=64/7, 16, 256/9, 256/5, 1024/11, 512/3, 496/13, ... =A084623(n+1)/A000265(n+1).]

			1;
2,1;
3,1,1;
5,1,1,1,1;
7,1,1,1,1,1,1;
11,1,1,1,1,1,1,1,1,1,1;
13,1,1,1,1,1,1,1,1,1,1,1,1;
17,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
19,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
23,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
29,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;

Cf. A076274 (row sums).

T(n,0) = A008578(n+1). T(n,m) =1, 1<=m<=A008578(n+1)-1.

cp's OEIS Frontend

A339393 Denominators of the probability that when a stick is broken up at n-1 points independently and uniformly chosen at random along its length there exist 3 of the n pieces that can form a triangle.

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A363399 Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (j + 1)^n), (tangent case).

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A301631 Numerator of population variance of n-th row of Pascal's triangle.

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A320086 Triangle read by rows, 0 <= k <= n: T(n,k) is the denominator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; numerator is A320085.

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A352485 Decimal expansion of the probability that when a unit interval is broken at two points uniformly and independently chosen at random along its length the lengths of the resulting three intervals are the altitudes of a triangle.

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A014964 a(n) = lcm(n, 2^(n-1)).

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A178623 Triangle T(n,m) read by rows: T(n,0)= prime(n); T(n,m)=1 if m>=1.

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