A103366 -id:A103366 - cp's OEIS frontend

A115369 Decimal expansion of first zero of BesselJ(1,z).

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

Offset: 1

Eric W. Weisstein, Jan 21 2006

Also the first root of the sinc(2,x) function, that is, the radial component of the 2D Fourier transform of a 2-dimensional unit disc. - Stanislav Sykora, Nov 14 2013

Also the first root of the derivative of BesselJ_0. - Jean-François Alcover, Jul 01 2015

			3.8317059702075123156...

Stanislav Sykora, K-Space Images of n-Dimensional Spheres and Generalized Sinc Functions
Eric Weisstein's World of Mathematics, Bessel Function Zeros
Index entries for transcendental numbers

Cf. A115368, A115370, A115371, A115372, A115373.

Cf. A103365, A103366, A238390, A180874.

BesselJZero[1, 1] // N[#, 105]& // RealDigits // First (* Jean-François Alcover, Feb 06 2013 *)

solve(x=3,4,besselj(1,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(1) \\ Charles R Greathouse IV, Aug 06 2022

A103365 First column of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).

Original entry on oeis.org

1, -1, 2, -7, 39, -321, 3681, -56197, 1102571, -27036487, 810263398, -29139230033, 1238451463261, -61408179368043, 3513348386222286, -229724924077987509, 17023649385410772579, -1419220037471837658603, 132236541042728184852942, -13690229149108218523467549

Offset: 1

Paul D. Hanna, Feb 02 2005

sign

			From _Paul D. Hanna_, Jan 31 2009: (Start)
G.f.: A(x) = 1 - x + 2*x^2/3 - 7*x^3/18 + 39*x^4/180 - 321*x^5/2700 +...
G.f.: A(x) = 1/B(x) where:
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 +...+ x^n/[n!*(n+1)!/2^n] +... (End)

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A001263, A103364, A103366, A103367.
Cf. A155926, A155927. [Paul D. Hanna, Jan 31 2009]
Cf. A238390, A180874, A115369.

Table[(-1)^((n-1)/2) * (CoefficientList[Series[x/BesselJ[1,2*x],{x,0,40}],x])[[n]] * ((n+1)/2)! * ((n-1)/2)!,{n,1,41,2}] (* Vaclav Kotesovec, Mar 01 2014 *)

a(n)=if(n<1,0,(matrix(n,n,m,j,binomial(m-1,j-1)*binomial(m,j-1)/j)^-1)[n,1])

{a(n)=local(B=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(1/B,n)*n!*(n+1)!/2^n} \\ Paul D. Hanna, Jan 31 2009

From Paul D. Hanna, Jan 31 2009: (Start)
G.f.: A(x) = 1/B(x) where A(x) = Sum_{n>=0} (-1)^n*a(n)*x^n/[n!*(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n].
G.f. satisfies: A(x) = 1/F(x*A(x)) and F(x) = 1/A(x*F(x)) where F(x) = Sum_{n>=0} A155926(n)*x^n/[n!*(n+1)!/2^n].
G.f. satisfies: A(x) = 1/G(x/A(x)) and G(x) = 1/A(x/G(x)) where G(x) = Sum_{n>=0} A155927(n)*x^n/[n!*(n+1)!/2^n]. (End)
a(n) ~ (-1)^(n+1) * c * n! * (n-1)! * d^n, where d = 4/BesselJZero[1, 1]^2 = 0.2724429913055159309179376055957891881897555639652..., and c = 9.11336321311226744479181866135367355200240221549667284076... = BesselJZero[1, 1]^2 / (4*BesselJ[2, BesselJZero[1, 1]]). - Vaclav Kotesovec, Mar 01 2014, updated Apr 01 2018

A103364 Matrix inverse of the Narayana triangle A001263.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -7, 12, -6, 1, 39, -70, 40, -10, 1, -321, 585, -350, 100, -15, 1, 3681, -6741, 4095, -1225, 210, -21, 1, -56197, 103068, -62916, 19110, -3430, 392, -28, 1, 1102571, -2023092, 1236816, -377496, 68796, -8232, 672, -36, 1, -27036487, 49615695, -30346380, 9276120, -1698732, 206388

Offset: 1

Paul D. Hanna, Feb 02 2005

The first column is A103365. The second column is A103366. Row sums are all zeros (for n > 1). Absolute row sums form A103367.

Let E(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = (1/sqrt(y))*BesselI(1,2*sqrt(y)). Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence n!*(n+1)! as defined in Wang and Wang. - Peter Bala, Aug 07 2013

			Rows begin:
        1;
       -1,        1;
        2,       -3,       1;
       -7,       12,      -6,       1;
       39,      -70,      40,     -10,     1;
     -321,      585,    -350,     100,   -15,     1;
     3681,    -6741,    4095,   -1225,   210,   -21,   1;
   -56197,   103068,  -62916,   19110, -3430,   392, -28,   1;
  1102571, -2023092, 1236816, -377496, 68796, -8232, 672, -36, 1;
  ...
From _Peter Bala_, Aug 09 2013: (Start)
The real zeros of the row polynomials R(n,x) appear to converge to zeros of E(alpha*x) as n increases, where alpha = -3.67049 26605 ... ( = -(A115369/2)^2).
Polynomial | Real zeros to 5 decimal places
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
R(5,x)     | 1, 3.57754, 3.81904
R(10,x)    | 1, 3.35230, 7.07532,  9.14395
R(15,x)    | 1, 3.35231, 7.04943, 12.09668, 15.96334
R(20,x)    | 1, 3.35231, 7.04943, 12.09107, 18.47845, 24.35255
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Function   |
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
E(alpha*x) | 1, 3.35231, 7.04943, 12.09107, 18.47720, 26.20778, ...
Note: The n-th zero of E(alpha*x) may be calculated in Maple 17 using the instruction evalf( BesselJZeros(1,n)/ BesselJZeros(1,1))^2 ). (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A000108, A001263, A055133, A086646, A103365, A103366, A103367, A104033.

T[n_, 1]:= Last[Table[(-1)^(n - 1)*(CoefficientList[Series[x/BesselJ[1, 2*x], {x, 0, 1000}], x])[[k]]*(n)!*(n - 1)!, {k, 1, 2*n - 1, 2}]]
T[n_, n_] := 1; T[2, 1] := -1; T[3, 1] := 2; T[n_, k_] := T[n, k] = T[n - 1, k - 1]*n*(n - 1)/(k*(k - 1)); Table[T[n, k], {n, 1, 50}, {k, 1, n}] // Flatten (* G. C. Greubel, Jan 04 2016 *)
T[n_, n_] := 1; T[n_, 1]/;n>1 := T[n, 1] = -Sum[T[n, j], {j, 2, n}]; T[n_, k_]/;1Oliver Seipel, Jan 01 2025 *)

PARI
```
T(n,k)=if(n
				
```

From Peter Bala, Aug 07 2013: (Start)
Let E(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = (1/sqrt(y))* BesselI(1,2*sqrt(y)). Generating function: E(x*y)/E(y) = 1 + (-1 + x)*y/(1!*2!) + (2 - 3*x + x^2)*y^2/(2!*3!) + (-7 + 12*x - 6*x^2 + x^3)*y^3/(3!*4!) + .... The n-th power of this array has a generating function E(x*y)/E(y)^n. In particular, the matrix inverse A001263 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} 1/(n-k+1)*binomial(n,k)*binomial(n+1,k+1) *R(k,x) with initial value R(0,x) = 1.
Let alpha denote the root of E(x) = 0 that is smallest in absolute magnitude. Numerically, alpha = -3.67049 26605 ... = -(A115369/2)^2. It appears that for arbitrary complex x we have lim_{n->oo} R(n,x)/R(n,0) = E(alpha*x). Cf. A055133, A086646 and A104033.
A stronger result than pointwise convergence may hold: the convergence may be uniform on compact subsets of the complex plane. This would explain the observation that the real zeros of the polynomials R(n,x) seem to converge to the zeros of E(alpha*x) as n increases. Some numerical examples are given below. (End)
From Werner Schulte, Jan 04 2017, corrected May 05 2025: (Start)
T(n,k) = T(n-1,k-1)*n*(n-1)/(k*(k-1)) for 1 < k <= n;
T(n,k) = T(n+1-k,1)*A001263(n,k) for 1 <= k <= n;
Sum_{k=1..n} T(n,k)*A000108(k) = 1 for n > 0. (End)

A103367 Absolute row sums of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).

Original entry on oeis.org

1, 2, 6, 26, 160, 1372, 15974, 245142, 4817712, 118198568, 3542890648, 127417949496, 5415490994368, 268526379444104, 15363229400769566, 1004545432884250126, 74441340170270921952, 6205992783298302119536

Offset: 2

Paul D. Hanna, Feb 02 2005

nonn

The ratio of the absolute row sum to the absolute value of column 1 for row n of triangle A103364, as n grows, approaches the limit given by limit_{n->inf} Sum_{k=1,n} |A103364(n,k)| / |A103365(n)| = 4.37281937295201487280058335227924496590808747668123198019676685492873...

Cf. A001263, A103364, A103365, A103366.

{a(n)=if(n<1,0,sum(k=1,n,abs(matrix(n,n,m,j,binomial(m-1,j-1)*binomial(m,j-1)/j)^-1)[n,k]))}

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A115369 Decimal expansion of first zero of BesselJ(1,z).

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A103365 First column of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).

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A103364 Matrix inverse of the Narayana triangle A001263.

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A103367 Absolute row sums of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).

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