A115368 -id:A115368 - cp's OEIS frontend

A000275 Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden.

1, 1, 3, 19, 211, 3651, 90921, 3081513, 136407699, 7642177651, 528579161353, 44237263696473, 4405990782649369, 515018848029036937, 69818743428262376523, 10865441556038181291819, 1923889742567310611949459, 384565973956329859109177427, 86180438505835750284241676121

Offset: 0

N. J. A. Sloane

a(n) has the Lucas property, namely a(n) is congruent to a(n_0)a(n_1)...a(n_k) modulo p for any prime p where n_0,n_1,... are the base p digits of n. (Carlitz via McIntosh)

			From _Peter Bala_, Aug 08 2011: (Start)
a(3) = 19: The 19 pairs of permutations in the group S_3 x S_3 with no common rises correspond to the zero entries in the table below.
  ======================================
   Number of common rises in S_3 x S_3
  ======================================
     | 123   132   213   231   312   321
  ======================================
  123|  2     1     1     1     1     0
  132|  1     1     0     1     0     0
  213|  1     0     1     0     1     0
  231|  1     1     0     1     0     0
  312|  1     0     1     0     1     0
  321|  0     0     0     0     0     0
(End)
G.f. = 1 + x + 3*x^2 + 19*x^3 + 211*x^4 + 3651*x^5 + 90921*x^6 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..261
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (8) (with t=0 and m=2) on p. 249.
Leonid Bedratyuk and Nataliia Luno, Connection problems for the generalized hypergeometric Appell polynomials, Carpathian Math. Publ. (2020) Vol. 12, No. 1, 10-18.
L. Carlitz, The coefficients of the reciprocal of J_0(x), Archiv. Math. 6 (1955), 121-127.
L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884.
L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884. [Annotated scanned copy]
L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975.
Jan Geuenich, Tilting modules for the Auslander algebra of K[x]/(xn), arXiv:1803.10707 [math.RT], 2018.
Gunnar Thor Magnússon, The inner product on exterior powers of a complex vector space, arXiv preprint arXiv:1401.4048 [math.AG], 2014.
R. McIntosh, A generalization of a congruential property of Lucas, Amer. Math. Monthly 99(3) (1992), 231-238; see page 232. MR1216210 (95b:11008)
J. Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498.
Jonathan D. H. Smith, Commutative Moufang loops and Bessel functions, Invent. Math. 67(1) (1982), 173-187.
Index entries for sequences related to Bessel functions or polynomials

Row 2 of A212855.
Cf. A055133 (absolute value of column 0 of triangle), A192721 (column 1), A115368.
Column k=1 of A340986.

A000275 := proc(n) sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): n!^2*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:
seq(A000275(n), n=0..17); # Peter Luschny, May 27 2017

a[0] = 1; a[n_] := a[n] = Sum[(-1)^(r+n+1)*Binomial[n, r]^2 a[r], {r, 0, n-1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 05 2013 *)
CoefficientList[Series[1/BesselJ[0,Sqrt[4*x]], {x, 0, 20}], x]* Range[0, 20]!^2 (* Vaclav Kotesovec, Mar 02 2014 *)
a[ n_] := If[ n < 0, 0, (n! 2^n)^2 SeriesCoefficient[ 1 / BesselJ[ 0, x], {x, 0, 2 n}]]; (* Michael Somos, Aug 20 2015 *)

{a(n) = if( n<0, 0, n!^2 * 4^n * polcoeff( 1 / besselj(0, x + x * O(x^(2*n))), 2*n))}; /* Michael Somos, May 17 2004 */

a(n) = Sum_{r=0..n-1} (-1)^(r+n+1) binomial(n, r)^2 a(r), if n > 0.
Sum_{n>=0} a(n) * x^n / n!^2 = 1 / J_0(sqrt(4*x)). - _Michael Somos, May 17 2004
From Peter Bala, Aug 08 2011: (Start)
Conjectural formula: 1 = Sum_{n>=0} a(n)*x^n*Sum_{k>=0} binomial(n+k,k)^2*(-x)^k.
Apart from the initial term, first column of A192721. (End)
E.g.f.: 1/J_0(sqrt(4*x)) = 1 + x/Q(0), where Q(k) = (k+1)^2 - x + (k+1)^2*x/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2013
a(n) ~ c * (n!)^2 / r^n, where r = (1/4)*BesselJZero[0,1]^2 = 1.4457964907366961302939989396139517587678604516... and c = 1/(sqrt(r) * BesselJ(1, 2*sqrt(r))) = 1.60197469692804662664846689139151227422675123376219... - Vaclav Kotesovec, Mar 02 2014, updated Apr 01 2018

More terms from Christian G. Bower, Apr 25 2000

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

Original entry on oeis.org

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

A002190 Sum_{n>=0} a(n)x^n/n!^2 = -log(BesselJ(0,2sqrt(x))).

Original entry on oeis.org

0, 1, 1, 4, 33, 456, 9460, 274800, 10643745, 530052880, 32995478376, 2510382661920, 229195817258100, 24730000147369440, 3113066087894608560, 452168671458789789504, 75059305956331837485345, 14121026957032156557396000, 2988687741694684876495689040

Offset: 0

N. J. A. Sloane

Number of non-ambiguous trees, see the Aval et al. reference. - Joerg Arndt, May 11 2015

			-log( Sum_{n>=0} (-x)^n/n!^2 ) = x + x^2/2!^2 + 4*x^3/3!^2 + 33*x^4/4!^2 + 456*x^5/5!^2 + 9460*x^6/6!^2 + ... . -_Paul D. Hanna_, Oct 09 2010

Stany De Smedt, On Sloane's Sequence 1484, Saitama Math. J. 15 (1997), 9-13.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..100
J.-C. Aval, A. Boussicault, M. Bouvel and M. Silimbani, Combinatorics of non-ambiguous trees, 2012. - From _N. J. A. Sloane_, Jan 03 2013
Jean-Christophe Aval, Adrien Boussicault, Mathilde Bouvel, Matteo, Combinatorics of non-ambiguous trees, arXiv:1305.3716 [math.CO], (16-May-2013).
Juan Arias de Reyna, Richard P. Brent and Jan van de Lune, On the sign of the real part of the Riemann zeta-function, arXiv preprint arXiv:1205.4423 [math.NT], 2012.
Beáta Bényi, Gábor V. Nagy, Bijective enumerations of Γ-free 0-1 matrices, arXiv:1707.06899 [math.CO], (2017).
L. Carlitz, A sequence of integers related to the Bessel functions, Proc. Amer. Math. Soc., 14 (1963), 1-9.
William Dugan, Sam Glennon, Paul E. Gunnells, Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], 2017.
Mark Dukes, Thomas Selig, Jason P. Smith, Einar Steingrimsson, Permutation graphs and the Abelian sandpile model, tiered trees and non-ambiguous binary trees, arXiv:1810.02437 [math.CO], 2018.
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO], 2006.
Christian Günther, Kai-Uwe Schmidt, Lq norms of Fekete and related polynomials, arXiv:1602.01750 [math.NT], 2016.
Joshua F. Robinson and Patrick B. Warren, Renormalisation group theory applied to x^.. + x^. + x^2 = 0, arXiv:2412.16578 [math-ph], 2024. See p. 5.
Index entries for sequences related to Bessel functions or polynomials

Cf. A101981. A diagonal of A217940.

Cf. A115368.

a:= n-> coeff(series(-ln(BesselJ(0,2*sqrt(x))), x, n+1), x, n)*(n!)^2:
seq(a(n), n=0..30); # Alois P. Heinz, Oct 10 2010

nn=18; CoefficientList[Series[-Log[BesselJ[0, 2*Sqrt[x]]], {x, 0, nn}], x]*Table[n!^2, {n, 0, nn}] (* Jean-François Alcover, Jun 22 2011 *)
Clear[q]; q[n_, 1] := (n-1)!^2; q[n_, k_] := q[n, k] = Sum[Binomial[n-1, j]*Binomial[n-1, j+1]*Sum[q[j+1, r]*q[n-j-1, k-r], {r, Max[1, -n+j+k+1], Min[j+1, k-1]}], { n-2}]; a[n_] := q[n, n]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 13 2013 *)

N=66; x='x+O('x^N);
f=-log(sum(n=0,N, (-x)^n/(n!)^2) );
f=serlaplace(f);
f=serlaplace(f);
concat([0],Vec(f))
\\ Joerg Arndt, May 17 2013

\\ Terms starting from a(1)=1:
N=33; B=vector(N);  B[1]=1;  b(j)=B[j+1];
for(n=0,N-2, B[n+2]=sum(i=0,n, my(j=n-i); binomial(n+1,i)*binomial(n+1,j)*b(i)*b(j) ) ); B
\\ Joerg Arndt, May 11 2015

Conjecture: G.f.: 1 = Sum_{n>=0} a(n+1)*A000108(n)*x^n*Sum_{k>=0} C(2*n+k,k)^2*(-x)^k. Compare with the following g.f of the Catalan numbers (A000108): 1 = Sum_{n>=0} A000108(n)*x^n*Sum_{k>=0} C(2*n+k,k)*(-x)^k. - Paul D. Hanna, Oct 10 2010
a(n) ~ n! * (n-1)! / r^n, where r = 1/4*BesselJZero[0,1]^2 = 1.44579649073669613... - Vaclav Kotesovec, Mar 02 2014
a(0) = 0; a(n) = -(-1)^n + (1/n) * Sum_{k=1..n-1} (-1)^(n-k-1) * binomial(n,k)^2 * k * a(k). - Ilya Gutkovskiy, Jul 15 2021

More terms and better definition from Vladeta Jovovic, Jul 16 2006

Edited by Assoc. Editors of the OEIS, Oct 12 2010

A340986 Square array read by descending antidiagonals. T(n,k) is the number of ways to separate the columns of an ordered pair of n-permutations (that have been written as a 2 X n array, one atop the other) into k cells so that no cell has a column rise. For n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 19, 0, 1, 4, 21, 92, 211, 0, 1, 5, 36, 255, 1354, 3651, 0, 1, 6, 55, 544, 4725, 29252, 90921, 0, 1, 7, 78, 995, 12196, 123903, 873964, 3081513, 0, 1, 8, 105, 1644, 26215, 377904, 4368729, 34555880, 136407699, 0

Offset: 0

Geoffrey Critzer, Feb 01 2021

A column rise (cf. A000275) means a pair of adjacent columns within a cell where each entry in the first column is less than the adjacent entry in the second column. The order of the columns cannot change. The cells are allowed to be empty.

			Square array T(n,k) begins:
  1,    1,     1,      1,      1,      1, ...
  0,    1,     2,      3,      4,      5, ...
  0,    3,    10,     21,     36,     55, ...
  0,   19,    92,    255,    544,    995, ...
  0,  211,  1354,   4725,  12196,  26215, ...
  0, 3651, 29252, 123903, 377904, 939155, ...

R. P. Stanley, Enumerative Combinatorics, Vol. I, Second Edition, Section 3.13.

Alois P. Heinz, Antidiagonals n = 0..100, flattened

Columns k=0-4 give: A000007, A000275, A336271, A336638, A336639.
Rows n=0-2 give: A000012, A001477, A014105.
Main diagonal gives A336665.
Cf. A212855, A192721, A334394.

T:= (n, k)-> n!^2*coeff(series(1/BesselJ(0, 2*sqrt(x))^k, x, n+1), x, n):
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Feb 02 2021

nn = 6; B[n_] := n!^2; e[x_] := Sum[x^n/B[n], {n, 0, nn}];
Table[Table[B[n], {n, 0, nn}] PadRight[CoefficientList[Series[e[-x]^-k, {x, 0, nn}], x], nn + 1], {k, 0, nn}] // Grid

Let E(x) = Sum_{n>=0} x^n/n!^2. Then Sum_{n>=0} T(n,k)*x^n/n!^2 = 1/E(-x)^k.
T(n,k) = (n!)^2 * [x^n] 1/BesselJ(0,2*sqrt(x))^k. - Alois P. Heinz, Feb 02 2021
For fixed k>=1, T(n,k) ~ 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 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 11 2025

A115370 Decimal expansion of first zero of BesselJ(2,z).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 21 2006

Also the first root of the sinc(4,x) function, that is, the radial component of the 4D Fourier transform of 4-dimensional unit sphere. Also, the solution of 2J_1(x) = x*J_0(x). - Stanislav Sykora, Nov 14 2013

			5.1356223018406825563...

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, A115369, A115371, A115372, A115373.

RealDigits[BesselJZero[2, 1], 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

solve(x=5,6,besselj(2,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(2) \\ Charles R Greathouse IV, Aug 06 2022

A115371 Decimal expansion of first zero of BesselJ(3,z).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 21 2006

			6.3801618959239835062...

Eric Weisstein's World of Mathematics, Bessel Function Zeros.
Index entries for transcendental numbers

Cf. A115368, A115369, A115370, A115372, A115373.

RealDigits[BesselJZero[3, 1], 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

solve(x=6,7,besselj(3,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(3) \\ Charles R Greathouse IV, Aug 06 2022

A115372 Decimal expansion of first zero of BesselJ(4,z).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 21 2006

			7.5883424345038043850...

Eric Weisstein's World of Mathematics, Bessel Function Zeros
Index entries for transcendental numbers

Cf. A115368, A115369, A115370, A115371, A115373.

RealDigits[BesselJZero[4,1],10,120][[1]] (* Harvey P. Dale, Jan 16 2012 *)

solve(x=7,8,besselj(4,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(4) \\ Charles R Greathouse IV, Aug 06 2022

A115373 Decimal expansion of first zero of BesselJ(5,z).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 21 2006

			8.7714838159599540191...

Eric Weisstein's World of Mathematics, Bessel Function Zeros.
Index entries for transcendental numbers

Cf. A115368, A115369, A115370, A115371, A115372.

RealDigits[BesselJZero[5, 1], 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

solve(x=8,9,besselj(5,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(5) \\ Charles R Greathouse IV, Aug 23 2022

A375451 Expansion of g.f. A(x) satisfying 0 = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * ([x^k] A(x)^n) for n >= 1.

Original entry on oeis.org

1, 1, 3, 21, 264, 5100, 138595, 5021209, 233863116, 13628372628, 972514037307, 83479400425677, 8490972592164813, 1010263560882000981, 139051185192340895094, 21926159523172792097194, 3927328317712845680689864, 793059545751159815604109176, 179339266160209677707004583560

Offset: 0

Paul D. Hanna, Sep 10 2024

nonn

Note that 0 = Sum_{k=0..n} (-1)^k * binomial(n,k) * ([x^k] G(x)^n) is satisfied by G(x) = 1/(1-x) for n >= 1.

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 264*x^4 + 5100*x^5 + 138595*x^6 + 5021209*x^7 + 233863116*x^8 + ...
The table of coefficients of x^k in A(x)^n begins:
  n=1: [1, 1,  3,  21,  264,  5100,  138595, ...];
  n=2: [1, 2,  7,  48,  579, 10854,  289415, ...];
  n=3: [1, 3, 12,  82,  954, 17352,  453657, ...];
  n=4: [1, 4, 18, 124, 1399, 24696,  632656, ...];
  n=5: [1, 5, 25, 175, 1925, 33001,  827900, ...];
  n=6: [1, 6, 33, 236, 2544, 42396, 1041046, ...];
  ...
from which we may illustrate the defining property given by
0 = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * ([x^k] A(x)^n).
Using the coefficients in the table above, we see that
  n=1: 0 = 1*1 - 1*1;
  n=2: 0 = 1*1 - 4*2 + 1*7;
  n=3: 0 = 1*1 - 9*3 + 9*12 - 1*82;
  n=4: 0 = 1*1 - 16*4 + 36*18 - 16*124 + 1*1399;
  n=5: 0 = 1*1 - 25*5 + 100*25 - 100*175 + 25*1925 - 1*33001;
  n=6: 0 = 1*1 - 36*6 + 225*33 - 400*236 + 225*2544 - 36*42396 + 1*1041046;
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A008459, A115368.

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = sum(k=0,#A-1, (-1)^(#A-k) * binomial(#A-1,k)^2 * polcoef(Ser(A)^(#A-1),k) )/(#A-1) ); A[n+1]}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * d^n * n!^2, where d = 0.691660276122579707675... = 4/BesselJZero(0,1)^2 = 4/A115368^2 and c = 3.8999463598998648630203... - Vaclav Kotesovec, Sep 10 2024

A181167 G.f.: 1 = Sum_{n>=0} a(n)x^n Sum_{k>=0} C(2n+k,k)^2*(-x)^k.

Original entry on oeis.org

1, 1, 8, 165, 6384, 397320, 36273600, 4566166605, 757975618400, 160424015864112, 42164387189608320, 13473505313334666600, 5144136790654611953280, 2312696796696904699224000, 1209297981696245764641077760, 727688337054213932985609546525

Offset: 0

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.

			E.g.f.: E(x) = 1 + x^2/2! + 8*x^4/4! + 165*x^6/6! + 6384*x^8/8! +...
where the e.g.f. equals the continued fraction:
E(x) = 1/(1 - x^2/(2 - x^2/(3 - x^2/(4 - x^2/(5 - x^2/(6 - x^2/(7 - x^2/(8 - x^2/(9 - x^2/(10 - ...)))))))))). [Due to Matthieu Josuat-Vergès]
Illustrate the g.f. by the series:
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 - 21^2*x^5 +...)
+ 8*x^2*(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 - 126^2*x^5 +...)
+ 165*x^3*(1 - 7^2*x + 28^2*x^2 - 84^2*x^3 + 210^2*x^4 - 462^2*x^5+...)
+ 6384*x^4*(1 - 9^2*x + 45^2*x^2 - 165^2*x^3 + 495^2*x^4 +...)
+ 397320*x^5*(1 - 11^2*x + 66^2*x^2 - 286^2*x^3 + 1001^2*x^4 +...)
+ 36273600*x^6*(1 - 13^2*x + 91^2*x^2 - 455^2*x^3 + 1820^2*x^4 +...)
+ 4566166605*x^7*(1 - 15^2*x + 120^2*x^2 - 680^2*x^3 + 3060^2*x^4 +...)
+...
Compare to a g.f. of the Catalan numbers (A000108):
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3*x + 6*x^2 - 10*x^3 + 15*x^4 - 21*x^5 +...)
+ 2*x^2*(1 - 5*x + 15*x^2 - 35*x^3 + 70*x^4 - 126*x^5 +...)
+ 5*x^3*(1 - 7*x + 28*x^2 - 84*x^3 + 210*x^4 - 462*x^5 +...)
+ 14*x^4*(1 - 9*x + 45*x^2 - 165*x^3 + 495*x^4 - 1287*x^5 +...)
+ 42*x^5*(1 - 11*x + 66*x^2 - 286*x^3 + 1001*x^4 - 3003*x^5 +...)
+ 132*x^6*(1 - 13*x + 91*x^2 - 455*x^3 + 1820*x^4 - 6188*x^5 +...)
+...
Surprisingly, terms a(n) are divisible by n*A000108(n) for n>0:
a(2)=2*2*2, a(3)=3*5*11, a(4)=4*14*114, a(5)=5*42*1892, a(6)=6*132*45800, a(7)=7*429*1520535, ..., a(n)=n*A000108(n)*A181168(n).

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Cf. A181168, A180716.
Cf. A002190. [From Paul D. Hanna, Oct 09 2010]
Cf. A115368.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, x/(y+1)*(b(x-1, y-1)+b(x-1, y+1))))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 08 2018

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]],{n,1,nmax}] (* Vaclav Kotesovec, Jul 31 2014 *)

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

/* Formula: a(n) = A000108(n)*A002190(n+1) implies: */
{a(n)=binomial(2*n,n)/(n+1)*(n+1)!^2*polcoeff(-log(sum(m=0,n+1,(-x)^m/m!^2)+O(x^(n+2))),n+1)} \\ Paul D. Hanna, Oct 09 2010

/* Continued Fraction expansion of the E.G.F.: */
{a(n)=local(CF=1+O(x));for(i=0,n,CF=1/((n-i+1)-x^2*CF));(2*n)!*polcoeff(CF,2*n)}

a(n) = n*A000108(n)*A181168(n) = C(2n,n-1)*A181168(n) for n>0, with a(0)=1.
a(n) = A000108(n)*A002190(n+1), where A002190 describes the coefficients in -log(BesselJ(0,2*sqrt(x))) and A000108 is the Catalan numbers. - Paul D. Hanna, Oct 09 2010
Differentiating -log(BesselJ(0,2*sqrt(x))) and substituting z=z^2 gives the e.g.f. Sum_{n>=0} a(n) * z^(2*n)/(2n)! = BesselJ(1,2*z)/z/BesselJ(0,2*z). Consequently, using Gauss' continued fraction, this e.g.f. is also: 1/(1-z^2/(2-z^2/(3-z^2/(4-z^2/(5-z^2/...))))). - Matthieu Josuat-Vergès, Apr 17 2011
E.g.f.: U(0) where U(k) = 1 - x^2/(x^2 - (k+1)*(k+2)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 15 2012
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 16/BesselJZero[0,1]^2 = 2.76664110449031883070186935..., c = 4/(sqrt(Pi)*BesselJZero[0,1]^2) = 0.390227523142124366836071453... . - Vaclav Kotesovec, Jul 31 2014

cp's OEIS Frontend

A000275 Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A002190 Sum_{n>=0} a(n)*x^n/n!^2 = -log(BesselJ(0,2*sqrt(x))).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A340986 Square array read by descending antidiagonals. T(n,k) is the number of ways to separate the columns of an ordered pair of n-permutations (that have been written as a 2 X n array, one atop the other) into k cells so that no cell has a column rise. For n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A115370 Decimal expansion of first zero of BesselJ(2,z).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A115371 Decimal expansion of first zero of BesselJ(3,z).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A002190 Sum_{n>=0} a(n)x^n/n!^2 = -log(BesselJ(0,2sqrt(x))).

A181167 G.f.: 1 = Sum_{n>=0} a(n)x^n Sum_{k>=0} C(2n+k,k)^2*(-x)^k.