A115365 -id:A115365 - cp's OEIS frontend

A243108 Decimal expansion of the 5th du Bois-Reymond constant.

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

Offset: 0

Vaclav Kotesovec, Aug 19 2014

The k-th Du Bois Reymond constant c_k is asymptotic to 2/(1+r^2)^(k/2), where r = A115365 = 4.4934094579090641753... is smallest positive root of the equation tan(r) = r. - Vaclav Kotesovec, Aug 20 2014

			0.00105610210733480920562199158210781176744608061...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239.

Vaclav Kotesovec, Table of n, a(n) for n = 0..450
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A062546 (c2), A224196 (c3), A207528 (c4), A245333 (c6).

Cf. A115365, A079330, A088989.

A055133 Matrix inverse of A008459 (squares of entries of Pascal's triangle).

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -19, 27, -9, 1, 211, -304, 108, -16, 1, -3651, 5275, -1900, 300, -25, 1, 90921, -131436, 47475, -7600, 675, -36, 1, -3081513, 4455129, -1610091, 258475, -23275, 1323, -49, 1, 136407699, -197216832, 71282064, -11449536, 1033900, -59584, 2352, -64, 1

Offset: 0

Christian G. Bower, Apr 25 2000

Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0,2*sqrt(-y)). Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - Peter Bala, Jul 24 2013

			Table T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
      1;
     -1,       1;
      3,      -4,     1;
    -19,      27,    -9,     1;
    211,    -304,   108,   -16,   1;
  -3651,    5275, -1900,   300, -25,   1;
  90921, -131436, 47475, -7600, 675, -36, 1;
  ... [edited by _Petros Hadjicostas_, Aug 24 2019]
From _Peter Bala_, Jul 24 2013: (Start)
Function   |        Real zeros to 5 decimal places
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
R(5,x)     | 1, 5.40649,  7.23983
R(10,x)    | 1, 5.26894, 12.97405, 18.53109
R(15,x)    | 1, 5.26894, 12.94909, 24.04769, 33.87883
R(20,x)    | 1, 5.26894, 12.94909, 24.04216, 38.54959, 53.32419
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
E(alpha*x) | 1, 5.26894, 12.94909, 24.04216, 38.54835, 56.46772, ...
where alpha = -1.44579 64907 ... ( = -(A115365/2)^2).
Note: The n-th zero of E(alpha*x) may be calculated in Maple 17 using the instruction evalf( (BesselJZeros(0,n)/BesselJZeros(0,1))^2 ). (End)

Alois P. Heinz, Rows n = 0..99, flattened
J. Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498; see p. 493 with beta_{n,k} = |T(n,k)|.
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, 308(24) (2008), 6466-6500.

Cf. A000275, A008459 (matrix inverse), A115365.

T:= proc(n) local M;
       M:= Matrix(n+1, (i, j)-> binomial(i-1, j-1)^2)^(-1);
       seq(M[n+1, i], i=1..n+1)
    end:
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 14 2013

T[n_] := Module[{M}, M = Table[Binomial[i-1, j-1]^2, {i, 1, n+1}, {j, 1, n+1}] // Inverse; Table[M[[n+1, i]], {i, 1, n+1}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Nov 28 2015, after Alois P. Heinz *)

T(n, k) = (-1)^(n+k)*A000275(n-k)*C(n, k)^2.
From Peter Bala, Jul 24 2013: (Start)
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0,2*sqrt(-y)). Generating function: E(x*y)/E(y) = 1 + (-1 + x)*y + (3 - 4*x + x^2)*y^2/2!^2 + (-19 + 27*x - 9*x^2 + x^3)*y^3/3!^2 + ....
The n-th power of this array has a generating function E(x*y)/E(y)^n. In particular, the matrix inverse A008459 has a generating function E(y)*E(x*y).
Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} binomial(n,k)^2*R(k,x) with initial value R(0,x) = 1.
There appears to be a connection between the zeros of the Bessel function E(x) and the real zeros of the row polynomials R(n,x). Let alpha denote the root of E(x) = 0 that is smallest in absolute magnitude. Numerically, alpha = -1.44579 64907 ... ( = -(A115365/2)^2). It appears that the real zeros of R(n,x) approach zeros of E(alpha*x) as n increases. A numerical example is given below. Indeed, it may be the case that lim_{n -> inf} R(n,x)/R(n,0) = E(alpha*x) for arbitrary complex x. (End)

A147862 Decimal expansion of smallest positive solution to x^2 = tan x.

Original entry on oeis.org

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

Offset: 1

Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008, Nov 19 2008

			4.66649956344464427694328301460179403021113626877285...

Index entries for transcendental numbers.

Cf. A115365, A147863, A147864, A147865, A147866, A147867, A147868.

RealDigits[x /. FindRoot[Tan[x] == x^2, {x, 4.5}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)

A147868 Decimal expansion of smallest positive solution to x^8 = tan x.

Original entry on oeis.org

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

Offset: 1

Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008

Index entries for transcendental numbers.

Cf. A115365, A147862-A147867.

RealDigits[x/.FindRoot[x^8==Tan[x],{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 04 2016 *)

A147864 Decimal expansion of smallest positive solution to x^4 = tan x.

Original entry on oeis.org

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

Offset: 1

Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008

Cf. A115365, A147862-A147868.

RealDigits[x/.FindRoot[x^4==Tan[x],{x,4.7,4.72}, WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Dec 15 2011 *)

A147867 Decimal expansion of smallest positive solution to x^7 = tan x.

Original entry on oeis.org

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

Offset: 1

Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008

			1.10211306720501101180409243264204757999956054042708...

Cf. A115365, A147862, A147863, A147864, A147865, A147866, A147868.

RealDigits[x /. FindRoot[Tan[x] == x^7, {x, 1}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)

A147863 Decimal expansion of smallest positive solution to x^3 = tan x.

Original entry on oeis.org

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

Offset: 1

Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008

			4.70277453705741362543379081104892669685697861675345...

Cf. A115365, A147862, A147864, A147865, A147866, A147867, A147868.

RealDigits[x /. FindRoot[Tan[x] == x^3, {x, 4.7}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)

A147865 Decimal expansion of smallest positive solution to x^5 = tan x.

Original entry on oeis.org

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

Offset: 1

Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008

			1.23181804314888978018243295767279236340151138139784...

Cf. A115365, A147862, A147863, A147864, A147866, A147867, A147868.

RealDigits[x /. FindRoot[Tan[x] == x^5, {x, 1.2}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)

A213053 Decimal expansion of the absolute minimum of sinc(x) = sin(x)/x (negated).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Jun 09 2012

Minimum value of the first negative lobe of sinc(x), attained for abs(x) = A115365.

The involute of the unit circle which starts at (1,0) crosses the x-axis for the first time at x = 1/a. - Álvar Ibeas, Jul 28 2017

			min[real x](sinc(x)) = -0.2172336282112216574082...

Cf. A115365.

digits = 105; NMinimize[ Sinc[x], x, WorkingPrecision -> digits+5] // First // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 05 2013 *)
RealDigits[Sinc[BesselJZero[3/2, 1]], 10, 100][[1]] (* Vladimir Reshetnikov, May 13 2016 *)

y=solve(x=4,4.5,tan(x)-x);-sin(y)/y \\ Charles R Greathouse IV, Jun 10 2012

-sinc(besseljzero(3/2,1)) \\ Charles R Greathouse IV, Jan 23 2025

Equals -1 / sqrt(1 + A115365^2) = cos(A115365). - Álvar Ibeas, Jul 28 2017

A377479 Decimal expansion of the smallest positive real root of the equation cos(x) + x*sin(x) = 0.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Oct 29 2024

The absolute value of the x-coordinate of the tangent point between the cosine graph and the straight line through the origin.

			2.79838604578388713672024890313957...

Index entries for transcendental numbers

Cf. A003957, A115365, A197002, A377480, A377481.

ndigits=100; First[RealDigits[First[x/.NSolve[Cos[x]+x Sin[x]==0,x,ndigits]],10,ndigits]]
(* or *)
RealDigits[BesselJZero[-3/2, 1], 10, 100][[1]] (* Vaclav Kotesovec, Oct 31 2024 *)

\\ Note: besseljzero not guaranteed to work here since -3/2 < 0.
solve(x=2,3, cos(x)+x*sin(x)) \\ Charles R Greathouse IV, Jan 23 2025

cp's OEIS Frontend

A243108 Decimal expansion of the 5th du Bois-Reymond constant.

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A055133 Matrix inverse of A008459 (squares of entries of Pascal's triangle).

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A147862 Decimal expansion of smallest positive solution to x^2 = tan x.

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A147868 Decimal expansion of smallest positive solution to x^8 = tan x.

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A147864 Decimal expansion of smallest positive solution to x^4 = tan x.

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A147867 Decimal expansion of smallest positive solution to x^7 = tan x.

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A147863 Decimal expansion of smallest positive solution to x^3 = tan x.

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A147865 Decimal expansion of smallest positive solution to x^5 = tan x.

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A213053 Decimal expansion of the absolute minimum of sinc(x) = sin(x)/x (negated).

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