A115368 -id:A115368 - cp's OEIS frontend

A336271 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * binomial(2k,k) a(n-k).

1, 2, 10, 92, 1354, 29252, 873964, 34555880, 1748176714, 110183215988, 8467704986260, 779536758060920, 84699429189141100, 10725613123706081720, 1565870044943751242440, 261092436660169105108592, 49312362996510562406915914, 10473104312824253527997052500

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Cf. A000275, A000984, A002893.

Column k=2 of A340986.

a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 Binomial[2 k, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^2.

a(n) ~ (n!)^2 * n / (BesselJ(1, 2*sqrt(r))^2 * r^(n+1)), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 15 2020

A181168 G.f.: 1 = 1/(1+x) + Sum_{n>=1} a(n)C(2n,n-1)x^n* Sum_{k>=0} C(2n+k,k)^2*(-x)^k.

Original entry on oeis.org

1, 2, 11, 114, 1892, 45800, 1520535, 66256610, 3666164264, 251038266192, 20835983387100, 2060833345614120, 239466622145739120, 32297762247056413536, 5003953730422122499023, 882564184814509784837250

Offset: 1

Paul D. Hanna, Oct 08 2010

nonn

Compare g.f. to a g.f of the Catalan numbers:

. 1 = Sum_{n>=0} A000108(n)*x^n * Sum_{k>=0} C(2n+k,k)*(-x)^k.

			Illustrate the g.f. by the series:
1 = (1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*1*1*x*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 +...)
+ 2*2*2*x^2*(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 +...)
+ 3*5*11*x^3*(1 - 7^2*x + 28^2*x^2 - 84^2*x^3 + 210^2*x^4 +...)
+ 4*14*114*x^4*(1 - 9^2*x + 45^2*x^2 - 165^2*x^3 + 495^2*x^4 +...)
+ 5*42*1892*x^5*(1 - 11^2*x + 66^2*x^2 - 286^2*x^3 + 1001^2*x^4 +...)
+ 6*132*45800*x^6*(1 - 13^2*x + 91^2*x^2 - 455^2*x^3 + 1820^2*x^4 +...)
+ 7*429*1520535*x^7*(1 - 15^2*x + 120^2*x^2 - 680^2*x^3 + 3060^2*x^4+..) +...
which indicates a connection of this sequence to the Catalan numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 1..260

Cf. A181167, A000108, A115368.

nmax=20; Table[(CoefficientList[Series[BesselJ[1,2*x]/x/BesselJ[0,2*x],{x,0,2*nmax}],x]*Range[0,2*nmax]!)[[2*n+1]] / Binomial[2n,n-1],{n,1,nmax}] (* Vaclav Kotesovec, Jul 31 2014 *)

{a(n)=if(n<1, 0, ((-1)^(n-1)-polcoeff(sum(m=0, n-1, a(m)*binomial(2*m, m-1)*x^m*sum(k=0, n-m, binomial(2*m+k, k)^2*(-x)^k)+x*O(x^n)), n))/binomial(2*n, n-1))}

a(n) = A181167(n)/C(2n,n-1) for n>=1.

a(n) ~ (n!)^2 * (2/BesselJZero[0,1])^(2*n+2), where BesselJZero[0,1] = A115368 = 2.40482555769... . - Vaclav Kotesovec, Jul 31 2014

A188489 Exponential transform of (A000275 number of pairs of permutations with rise/rise forbidden).

Original entry on oeis.org

1, 1, 2, 8, 61, 797, 16021, 457285, 17529203, 867230231, 53745914922, 4076301322848, 371301496685164, 39992538951200636, 5027440719872343598, 729432303460596468394, 120977789712983152108734, 22743262423568258626295550

Offset: 0

Paul D. Hanna, Apr 01 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 61*x^4 + 797*x^5 + 16021*x^6 +...
log(A(x)) = x + 3*x^2/2 + 19*x^3/3 + 211*x^4/4 + 3651*x^5/5 + 90921*x^6/6 +...+ A000275(n)*x^n/n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A000275 (log), A115368.

{A000275(n)=n!^2*4^n*polcoeff(1/besselj(0, x+x*O(x^(2*n))), 2*n)}
{a(n)=polcoeff(exp(sum(m=1,n,A000275(m)*x^m/m)+x*O(x^n)),n)}

G.f.: A(x) = exp( Sum_{n>=1} A000275(n)*x^n/n ) where A000275 is the number of pairs of permutations with rise/rise forbidden.

a(n) ~ c * n! * (n-1)! / r^n, where r = 1/4*BesselJZero[0,1]^2 = 1.44579649073669613 and c = 1/(sqrt(r) * BesselJ(1, 2*sqrt(r))) = 1.6019746969280466266484... - Vaclav Kotesovec, Mar 02 2014, updated Apr 01 2018

A244354 Decimal expansion of 'mu', a Sobolev isoperimetric constant related to the "membrane inequality", arising from the study of a vibrating membrane that is stretched across the unit disk and fastened at its boundary.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 26 2014

			0.17291506903064492691886683443...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.6 Sobolev Isoperimetric Constants, p. 221.

Eric Weisstein's MathWorld, Bessel Function Zeros
Index entries for transcendental numbers

Cf. A115368.

theta = BesselJZero[0, 1]; mu = 1/theta^2; RealDigits[mu, 10, 104] // First

1/besseljzero(0)^2 \\ Charles R Greathouse IV, Aug 23 2022

mu = 1/theta^2, where theta is A115368, the first positive zero of the Bessel function J0(x).

A336638 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^3.

Original entry on oeis.org

1, 3, 21, 255, 4725, 123903, 4368729, 199467243, 11455187445, 808475761695, 68805857523321, 6950458374996843, 822292004658568761, 112639503374757412875, 17688916392275574761805, 3157133540377493872350855, 635546443798928578953138165

Offset: 0

Ilya Gutkovskiy, Jul 28 2020

nonn

Cf. A000275, A002893, A336271, A336639.

Column k=3 of A340986.

nmax = 16; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^3, {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 HypergeometricPFQ[{1/2, -k, -k}, {1, 1}, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * A002893(k) * a(n-k).

a(n) ~ n!^2 * n^2 / (2 * r^(n + 3/2) * BesselJ(1, 2*sqrt(r))^3), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 11 2025

A336639 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^4.

Original entry on oeis.org

1, 4, 36, 544, 12196, 377904, 15438816, 803602944, 51908768676, 4074743122384, 382079412133936, 42184889139337344, 5417567866536188896, 800808722921088352384, 135006904500993157933056, 25751088570167886107910144, 5517695042810314282550432676

Offset: 0

Ilya Gutkovskiy, Jul 28 2020

nonn

In general, if k>=1 and Sum_{n>=0} a(n) * x^n / n!^2 = 1 / BesselJ(0, 2*sqrt(x))^k, then a(n) ~ n!^2 * n^(k-1) / ((k-1)! * r^(n + k/2) * BesselJ(1, 2*sqrt(r))^k), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4. - Vaclav Kotesovec, Jul 11 2025

Cf. A000275, A002895, A336271, A336638.

Column k=4 of A340986.

nmax = 16; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^4, {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 Binomial[2 k, k] HypergeometricPFQ[{1/2, -k, -k, -k}, {1, 1, 1/2 - k}, 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * A002895(k) * a(n-k).

a(n) ~ n!^2 * n^3 / (6 * r^(n+2) * BesselJ(1, 2*sqrt(r))^4), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 11 2025

A244355 Decimal expansion of 'lambda', a Sobolev isoperimetric constant related to the "membrane inequality", arising from the study of a vibrating membrane that is stretched across the unit disk and fastened at its boundary.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 26 2014

			5.7831859629467845211759957584558...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.6 Sobolev Isoperimetric Constants, p. 221.

Robert Stephen Jones, The fundamental Laplacian eigenvalue of the regular polygon with Dirichlet boundary conditions, arXiv:1712.06082 [math.NA], 2017, p. 17.
Eric Weisstein's MathWorld, Bessel Function Zeros
Index entries for transcendental numbers

Cf. A115368, A244354.

theta = BesselJZero[0, 1]; lambda = theta^2; RealDigits[lambda, 10, 103] // First

solve(x=2, 3, besselj(0, x))^2 \\ Michel Marcus, Nov 02 2018

besseljzero(0)^2 \\ Charles R Greathouse IV, Aug 09 2022

lambda = theta^2 where theta is A115368, the first positive zero of the Bessel function J0(x).
lambda = 1/mu = 1/A244354.
lambda is also the smallest eigenvalue of the ODE r^2*g''(r)+r*g'(r)+lambda*r^2*g(r)=0, g(0)=1, g(1)=0.

A321215 Decimal expansion of C[11] coefficient (negated) in 1/N expansion of lowest Laplacian Dirichlet eigenvalue of the Pi-area, N-sided regular polygon.

Original entry on oeis.org

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

Offset: 4

Robert Stephen Jones, Oct 31 2018

This is the 11th coefficient C[11] = -6016.337... in the 1/N expansion of the lowest Laplacian Dirichlet eigenvalue of the N-sided, Pi-area regular polygon.
In context, the eigenvalue expression for the N-sided, Pi-area regular polygon is L = L0*(1 + C[3]/N^3 + C[5]/N^5 + C[6]/N^6 + C[7]/N^7 + C[8]/N^8 + ... + C[11]/N^11 + ...) where L0 = [A115368]^2 = [A244355] is the eigenvalue in the Pi-area circle.
C[11] was computed by first computing several hundred 200-digit eigenvalues in the range from N = 1000 to 3000, and then using linear regression to determine this expansion coefficient. All digits reported are correct. This is the first coefficient that appears to break the regular pattern involving roots of Bessel functions and Riemann zeta functions, for example, C[3] = 4*zeta(3) and C[5] = (12-2*L0)*zeta(5), where zeta(n) is the Riemann zeta function. C[11] is negative.

			6016.335717690346829221853315075454811530972180617310177993314476104546100896...

Robert Stephen Jones, Table of n, a(n) for n = 4..132 (sign corrected by _Georg Fischer_, Jan 20 2019)
Mark Boady, Applications of Symbolic Computation to the Calculus of Moving Surfaces. PhD thesis, Drexel University, Philadelphia, PA. 2015.
P. Grinfeld and G. Strang, Laplace eigenvalues on regular polygons: A series in 1/N, J. Math. Anal. Appl., 385-149, 2012.
Robert Stephen Jones, Computing ultra-precise eigenvalues of the Laplacian within polygons. Advances in Computational Mathematics, May 2017.
Robert Stephen Jones, The fundamental Laplacian eigenvalue of the regular polygon with Dirichlet boundary conditions, arXiv:1712.06082 [math.NA], 2017.

Cf. A321216 = C[12], the next coefficient in the 1/N expansion.

Cf. A115368, A244355, A002117, and A013663.

A321216 Decimal expansion of C[12] coefficient in 1/N expansion of the lowest Laplacian Dirichlet eigenvalue of the Pi-area, N-sided regular polygon.

Original entry on oeis.org

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

Offset: 5

Robert Stephen Jones, Oct 31 2018

This is the 12th coefficient C[12] in the 1/N expansion of the lowest Laplacian Dirichlet eigenvalue of Pi-area, N-sided regular polygon. It was determined using experimental mathematics by computing the coefficient to 125 digits of precision. It can be computed using the expression in the Formula section. It is expressed in terms of L0 = [A115368]^2 = [A244355] = 5.78318... (eigenvalue of unit-radius circle) and Riemann zeta functions. Although this is derived using experimental mathematics, the decimal expansion reported is equal to that expression. In context, the eigenvalue expression for the N-sided, Pi-area regular polygon is

L = L0*(1 + C[3]/N^3 + C[5]/N^5 + C[6]/N^6 + C[7]/N^7 + C[8]/N^8 + ... + C[12]/N^12 + ...). The expression for this coefficient follows a pattern similar to lower-order coefficients (except C[11] [A321215]), e.g., C[3]=4*zeta(3) and C[5]=(12-2*L0)*zeta(5).

			25200.9737929324646760652122395385477028780653225566146497901539447736054240...

Robert Stephen Jones, Table of n, a(n) for n = 5..1004
Mark Boady, Applications of Symbolic Computation to the Calculus of Moving Surfaces. PhD thesis, Drexel University, Philadelphia, PA. 2015.
P. Grinfeld and G. Strang, Laplace eigenvalues on regular polygons: A series in 1/N, J. Math. Anal. Appl., 385-149, 2012.
Robert Stephen Jones, The fundamental Laplacian eigenvalue of the regular polygon with Dirichlet boundary conditions, arXiv:1712.06082 [math.NA], 2017.
Robert Stephen Jones, Computing ultra-precise eigenvalues of the Laplacian within polygons, Advances in Computational Mathematics, May 2017.

Cf. A321215 is decimal expansion of C[11], the next lower order coefficient.

Cf. A115368, A244355, A002117, and A013663.

{default(realprecision,100);L0=solve(x=2,3,besselj(0,x))^2;(32/3+272*L0/3-16*L0^2)*zeta(3)^4+(1360/3-488*L0/3+456*L0^2+107*L0^3/3+5*L0^4/8)*zeta(3)*zeta(9)+(432-216*L0-207*L0^2+47*L0^3/2+11*L0^4/8)*zeta(5)*zeta(7)}

A242402 Decimal expansion of the smallest positive root of the equation J_0(t)I_1(t)+I_0(t)J_1(t) = 0 (with I_0, I_1, J_0 and J_1, Bessel functions).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 13 2014

"This [constant] is associated with the study of a vibrating, homogeneous plate clamped at the boundary [of the unit disk]." - Quoted from Steven R. Finch.

			3.196220616582541093980527403403720341599...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 222.

Cf. A115368.

FindRoot[BesselJ[0, t]*BesselI[1, t] + BesselI[0, t]*BesselJ[1, t] == 0, {t, 3}, WorkingPrecision -> 100][[1, 2]] // RealDigits[#, 10, 100]& // First

solve(t=3,4, besselj(0,t)*besseli(1,t)+besseli(0,t)*besselj(1,t)) \\ Charles R Greathouse IV, Oct 23 2023

cp's OEIS Frontend

A336271 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * binomial(2*k,k) * a(n-k).

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A181168 G.f.: 1 = 1/(1+x) + Sum_{n>=1} a(n)*C(2n,n-1)*x^n* Sum_{k>=0} C(2n+k,k)^2*(-x)^k.

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A188489 Exponential transform of (A000275 number of pairs of permutations with rise/rise forbidden).

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A244354 Decimal expansion of 'mu', a Sobolev isoperimetric constant related to the "membrane inequality", arising from the study of a vibrating membrane that is stretched across the unit disk and fastened at its boundary.

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A336638 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^3.

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A336639 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^4.

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A244355 Decimal expansion of 'lambda', a Sobolev isoperimetric constant related to the "membrane inequality", arising from the study of a vibrating membrane that is stretched across the unit disk and fastened at its boundary.

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A321215 Decimal expansion of C[11] coefficient (negated) in 1/N expansion of lowest Laplacian Dirichlet eigenvalue of the Pi-area, N-sided regular polygon.

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A336271 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * binomial(2k,k) a(n-k).

A181168 G.f.: 1 = 1/(1+x) + Sum_{n>=1} a(n)C(2n,n-1)x^n* Sum_{k>=0} C(2n+k,k)^2*(-x)^k.

A242402 Decimal expansion of the smallest positive root of the equation J_0(t)I_1(t)+I_0(t)J_1(t) = 0 (with I_0, I_1, J_0 and J_1, Bessel functions).