A002474 -id:A002474 - cp's OEIS frontend

A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

1, 4, 64, 2304, 147456, 14745600, 2123366400, 416179814400, 106542032486400, 34519618525593600, 13807847410237440000, 6682998146554920960000, 3849406932415634472960000, 2602199086312968903720960000, 2040124083669367620517232640000, 1836111675302430858465509376000000

Offset: 0

N. J. A. Sloane

Denominators in the series for Bessel's J0(x) = 1 - x^2/4 + x^4/64 - x^6/2304 + ...
a(n) is the unreduced numerator in Product_{k=1..n} (4*k^2)/(4*k^2-1), therefore a(n)/A079484(n) = Pi/2 as n -> oo. - Daniel Suteu, Dec 02 2016
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture: Let zeta be a primitive 2n+1-th root of unity. Then the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} is a(n)/(2n+1) = ((2n)!!)^2/(2n+1), where m(j,k) is 1 or (1+zeta^(j-k))/(1-zeta^(j-k)) according as j = k or not.
The determinant of the matrix [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n)!!)^2/(2n(2n+1)) by Han Wang and Zhi-Wei Sun in 2022. (End)

Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (20.1).
Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.4.7
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 110.
E. L. Ince, Ordinary Differential Equations, Dover, NY, 1956; see p. 173.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 49 and 52, equations 49:6:1 and 52:6:2 at pages 483, 513.

T. D. Noe, Table of n, a(n) for n = 0..50
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.
Han Wang and Zhi-Wei Sun, Proof of a conjecture involving derangements and roots of unity, arXiv:2206.02589 [math.CO], 2022.
Index to divisibility sequences.
Index entries for sequences related to factorial numbers.

Cf. A000165, A001818, A079484, A197036, A334380.

J1: A002474, J2: A002506, J3: A014401.

[4^n*Factorial(n)^2: n in [0..15]]; // Vincenzo Librandi, Mar 15 2019

Array[4^# (#!)^2 &, 14, 0] (* Michael De Vlieger, Nov 01 2017 *)

a(n) = 4^n*(n!)^2; \\ Michel Marcus, Mar 13 2019

(-1)^n*a(n) is the coefficient of x^1 in Product_{k=0..2*n} (x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002
E.g.f.: A(x) = arcsin(x)*sec(arcsin(x)). - Vladimir Kruchinin, Sep 12 2010
E.g.f.: arcsin(x)*sec(arcsin(x)) = arcsin(x)/sqrt(1-x^2) = x/G(0); G(k) = 2k*(x^2+1)+1-x^2*(2k+1)*(2k+2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+2)^2/(1-x/(x - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
From Ilya Gutkovskiy, Dec 02 2016: (Start)
a(n) ~ Pi*2^(2*n+1)*n^(2*n+1)/exp(2*n).
Sum_{n>=0} 1/a(n) = BesselI(0,1) = A197036. (End)
From Daniel Suteu, Dec 02 2016: (Start)
a(n) ~ 2^(2*n) * gamma(n+1/2) * gamma(n+3/2).
a(n) ~ Pi*(2*n+1)*(4*n^2-1)^n/exp(2*n). (End)
2*a(n)/(2*n+1)! = A101926(n) / A001803(n). - Daniel Suteu, Feb 03 2017
Limit_{n->oo} n*a(n)/((2n+1)!!)^2 = Pi/4. - Daniel Suteu, Nov 01 2017
Sum_{n>=0} (-1)^n/a(n) = BesselJ(0, 1) (A334380). - Amiram Eldar, Apr 09 2022
Limit_{n->oo} a(n) / (n * A001818(n)) = Pi. - Daniel Suteu, Apr 09 2022

A002506 Denominators of coefficients of expansion of Bessel function J_2(x).

Original entry on oeis.org

8, 96, 3072, 184320, 17694720, 2477260800, 475634073600, 119859786547200, 38355131695104000, 15188632151261184000, 7290543432605368320000, 4170190843450270679040000, 2802368246798581896314880000

Offset: 0

N. J. A. Sloane

nonn

			a(2) = 3072 = 64*2*24, J2(x) = x^2/8 - x^4/96 + x^6/3072 - x^8/184320 +- ...

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.4.7

J0: A002454, J1: A002474, J3: A014401.

Denominator[Take[CoefficientList[Series[BesselJ[2,x],{x,0,30}],x],{3,-1,2}]] (* Harvey P. Dale, Sep 21 2013 *)

a(n) = 2^(2n+k) * n! * (n+k)! here for k=2, i.e., Bessel's J2(x).

a(n) - 4*n*(n+2)*a(n-1) = 0. - R. J. Mathar, Jun 20 2013

Previous Mathematica program corrected by Harvey P. Dale, Sep 21 2013

A014401 Denominators of coefficients of expansion of Bessel function J_3(x).

Original entry on oeis.org

48, 768, 30720, 2211840, 247726080, 39636172800, 8561413324800, 2397195730944000, 843812897292288000, 364527171630268416000, 189554129247739576320000, 116765343616607579013120000

Offset: 0

N. J. A. Sloane

nonn

			a(1) = 768 = 32*24, J3(x) = x^3/48 - x^5/768 + x^7/30720 - x^9/2211840 +- ...

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.4.7

J0: A002454, J1: A002474, J2: A002506.

k:=3: f:= gfun:-rectoproc({a(n)-(4*n^2 + 4*n*k)*a(n-1), a(0)=2^k*k!}, a(n), remember): map(f, [$0..16]); # Georg Fischer, Mar 22 2022

Denominator[Take[CoefficientList[Series[BesselJ[3,x],{x,0,30}],x],{4,-1,2}]] (* Harvey P. Dale, Dec 10 2011 *)

a(n) = 2^(2n+k) * n! * (n+k)! here for k=3, i.e., Bessel's J3(x).

D-finite with recurrence: a(n) - (4*n^2 + 4*n*k)*a(n-1) = 0, a(0) = 2^k*k!, here for k=3. - Georg Fischer, Mar 22 2022

A001900 Successive numerators of Wallis's approximation to Pi/2 (unreduced).

Original entry on oeis.org

1, 2, 4, 16, 64, 384, 2304, 18432, 147456, 1474560, 14745600, 176947200, 2123366400, 29727129600, 416179814400, 6658877030400, 106542032486400, 1917756584755200, 34519618525593600, 690392370511872000, 13807847410237440000, 303772643025223680000

Offset: 0

N. J. A. Sloane

a(n) = number of permutations of [n+1] all of whose non-initial left-to-right minima are at even positions in the permutation. For example, a(2) = 4 counts 123, 132, 213, 312. - David Callan, Jul 22 2008
Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal. a(2) = 4: [(0,0),(1,0),(2,0)], [(0,0),(0,1),(1,0),(2,0)], [(0,0),(0,1),(0,2),(1,1),(2,0)], [(0,0),(1,0),(0,1),(0,2),(1,1),(2,0)]. - Alois P. Heinz, Mar 23 2017
a(n+1) is the number of 0-1 square matrices of order n+1 with 2n+1 nonzero entries where the cell (i,j) is 1 for all i+j=n+2 and every diagonal, parallel to the main diagonal, has exactly one 1. For example, a(2) = 4: [(0,1,1), (1,1,0), (1,0,0)], [(0,1,1), (0,1,0), (1,1,0)], [(0,0,1), (1,1,1), (1,0,0)], [(0,0,1), (0,1,1), (1,1,0)]. - Christian Barrientos, Jul 17 2021

			From _Wolfdieter Lang_, Dec 06 2017: (Start)
Partial products of the rows N (for numerators a(n)) and D (for denominators b(n) = A000246(n+1)) begin:
n:    0  1  2  3  4   5    6     7      8       9       10 ...
N:    1  2  2  4  4   6    6     8      8      10       10 ...
D:    1  1  3  3  5   5    7     7      9       9       11 ...
a(n): 1  2  4 16 64 384 2304 18432 147456 14745601 4745600 ...
b(n): 1  1  3  9 45 225 1575 11025  99225   893025 9823275 ...  (End)

H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.

Alois P. Heinz, Table of n, a(n) for n = 0..449
John Derbyshire, Prime Obsession, Plume books, p. 16, 2003.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
Index to divisibility sequences

Cf. A000246, A284230.

For the reduced form see A001901(n)/A001902(n), n >= 0.

a[n_] := a[n] = If[n==0, 1, (n+Mod[n, 2]) a[n-1]];
a /@ Range[0, 21] (* Jean-François Alcover, Jan 31 2020 *)

a(n)=if(n<0,0,prod(k=1,n,if(k%2,k+1,k)))

2.2.4.4.6.6....2n.2n.../1.3.3.5.5.7.7....(2n-1).(2n+1) ...for n >= 1.
a(n) = 2^n * A010551(n) = 2^n * [n/2]! * [(n+1)/2]!. - Ralf Stephan, Mar 11 2004
Conjecture: a(n) - a(n-1) - n*(n-1)*a(n-2) = 0. - R. J. Mathar, Jun 07 2013 [The proof, for n >= 2, follows from the bisection recurrence given below. - Wolfdieter Lang, Dec 07 2017]
E.g.f.: E(0), where E(k)= 1 + 2*x*(k+1)/((2*k+1) - x*(2*k+1)/(x + 1/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 08 2013
G.f.: G(0), where G(k)= 1 + 2*x*(k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 08 2013
Bisection: a(2*k+1) = ((2*k+1)+1)*a(2*k), a(2*k) = 2*k*a(2*k-1), k >= 0, with a(0) = 1. The proof is obvious from the numbers in the numerator (see the row N in the example). From a proposal by David James Sycamore, Nov 02 2017 based on the fractions 4/1, 8/3, 32/9, 128/45, ... converging very slowly to Pi, given on p. 16 of the Derbyshire link. - Wolfdieter Lang, Dec 06 2017
From Mathias Zechmeister, Jul 24 2022: (Start)
Let J_0(x) and J_1(x) denote the Bessel functions and i = sqrt(-1).
a(n) = denominator([x^n](J_0(x) + J_1(x))).
a(n) = denominator([x^n](J_0(i*x) - i*J_1(i*x))).
Bisection: a(2*k) = A002454(k), a(2*k+1) = A002474(k).
G.f. for 1/a(n): J_0(i*x) - i*J_1(i*x).  (End)

A161736 Denominators of the column sums of the BG2 matrix.

Original entry on oeis.org

1, 9, 75, 1225, 19845, 160083, 1288287, 41409225, 1329696225, 10667118605, 85530896451, 1371086188563, 21972535073125, 176021737014375, 1409850293610375, 90324408810638025, 5786075364399106425, 46326420401234675625, 370882277949065911875, 5938020471163465810125

Offset: 2

Johannes W. Meijer, Jun 18 2009

The BG2 matrix coefficients, see also A008956, are defined by BG2[2m,1] = 2*beta(2m+1) and the recurrence relation BG2[2m,n] = BG2[2m,n-1] - BG2[2m-2,n-1]/(2*n-3)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). We observe that beta(2m+1) = 0 for m = -1, -2, -3, .. .
A different way to define the matrix coefficients is BG2[2*m,n] = (1/m)*sum(LAMBDA(2*m-2*k,n-1)*BG2[2*k,n], k=0..m-1) with LAMBDA(2*m,n-1) = (1-2^(-2*m))*zeta(2*m)-sum((2*k-1)^(-2*m), k=1..n-1) and BG2[0,n] = Pi/2 for m = 0, 1, 2, .. , and n = 1, 2, 3 .. , with zeta(m) the Riemann zeta function.
The columns sums of the BG2 matrix are defined by sb(n) = sum(BG2[2*m,n], m=0..infinity) for n = 2, 3, .. . For large values of n the value of sb(n) approaches Pi/2.
It is remarkable that if we assume that BG2[2m,1] = 2 for m = 0, 1, .. the columns sums of the modified matrix converge to the original sb(n) values. The first Maple program makes use of this phenomenon and links the sb(n) with the central factorial numbers A008956.
The column sums sb(n) can be linked to other sequences, see the second Maple program.
We observe that the column sums sb(n) of the BG2(n) matrix are related to the column sums sl(n) of the LG2(n) matrix, see A008956, by sb(n) = (-1)^(n+1)*(2*n-1)*sl(n).
a(n+2), for n >= 0, seems to coincide with the numerators belonging to A278145. - Wolfdieter Lang, Nov 16 2016
Suppose that, given values f(x-2*n+1), f(x-2*n+3), ..., f(x-1), f(x+1), ..., f(x+2*n-3), f(x+2*n-1), we approximate f(x) using the first 2*n terms of its Taylor series. Then 1/sb(n+1) is the coefficient of f(x-1) and f(x+1). - Matthew House, Dec 03 2024

			sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..

G. C. Greubel, Table of n, a(n) for n = 2..830
R. Arratia, S. Garibaldi, and J. Kilian, Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process, arXiv:1310.7055 [math.PR], 2013.

Cf. Numerators A161737.
Cf. A001818, A001902, A008956, A013777, A134372 and A134375.
Cf. A001147, A133221, A161738 and A002474, A278145.

[Denominator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // G. C. Greubel, Sep 26 2018

nmax := 18; for n from 0 to nmax do A001818(n) := (doublefactorial(2*n-1))^2 od: for n from 0 to nmax do A008956(n, 0):=1 od: for n from 0 to nmax do A008956(n, n) := A001818(n) od: for n from 1 to nmax do for m from 1 to n-1 do A008956(n, m) := (2*n-1)^2*A008956(n-1, m-1) + A008956(n-1, m) od: od: for n from 1 to nmax do for m from 0 to n do s(n, m):=0; s(n, m) := s(n, m)+ sum((-1)^k1*A008956(n, n-k1), k1=0..n-m): od: sb1(n+1) := sum(s(n, k1), k1=1..n) * 2/A001818(n); od: seq(sb1(n), n=2..nmax); # End program 1
nmax1 := nmax; for n from 0 to nmax1 do A001147(n):= doublefactorial(2*n-1) od: for n from 0 to nmax1/2 do A133221(2*n+1) := A001147(n); A133221(2*n) := A001147(n) od: for n from 0 to nmax1 do A002474(n) := 2^(2*n+1)*n!*(n+1)! od: for n from 1 to nmax1 do A161738(n) := ((product((2*n-3-2*k1), k1=0..floor(n/2-1)))) od: for n from 2 to nmax1 do sb2(n) := A002474(n-2) / (A161738(n)*A133221(n-1))^2 od: seq(sb2(n), n=2..nmax1); # End program 2
# Above Maple programs edited by Johannes W. Meijer, Sep 25 2012
r := n -> (1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> numer(simplify(r(n))):
seq(a(n), n = 1..21);  # Peter Luschny, Feb 12 2025

sb[2]=2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2; Table[sb[n] // Denominator, {n, 2, 20}] (* Jean-François Alcover, Aug 14 2017 *)

{a(n) = if( n<2, 0, n--; numerator( binomial( 2*n, n)^2 * n / 2^(n+1) ))}; /* Michael Somos, May 09 2011 */

a(n) = denom(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161737(n) = numer(sb(n)).
a(n+1) = numerator of C(2*n,n)^2 * n / 2^(n+1). - Michael Somos, May 09 2011
a(n) = A001902(2*n-3). - Mats Granvik, Nov 25 2018
a(n) = numerator((1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - Peter Luschny, Feb 13 2025

A160481 Row sums of the Beta triangle A160480.

Original entry on oeis.org

-1, -10, -264, -13392, -1111680, -137030400, -23500108800, -5351202662400, -1562069156659200, -568747270103040000, -252681700853514240000, -134539938778433126400000, -84573370199475510312960000, -61972704966344777143418880000, -52361960516341326660973363200000

Offset: 2

Johannes W. Meijer, May 24 2009, Sep 19 2012

It is conjectured that the row sums of the Beta triangle depend on three different sequences. Two Maple algorithms are given. The first one gives the row sums according to the Beta triangle A160480 and the second one gives the row sums according to our conjecture.

Christopher P. Herzog, Kuo-Wei Huang, and Kristan Jensen, Universal Entanglement and Boundary Geometry in Conformal Field Theory, arXiv preprint arXiv:1510.00021 [hep-th], 2015.
Kuo-Wei Huang, Resummation of Multi-Stress Tensors in Higher Dimensions, arXiv:2406.07458 [hep-th], 2024. See p. 10.

A160480 is the Beta triangle.
Row sum factors A120778, A000165 and A049606.
Cf. A002474, A009445, A129890.

nmax := 14; mmax := nmax: for n from 1 to nmax do BETA(n, n) := 0 end do: m := 1: for n from m+1 to nmax do BETA(n,m) := (2*n-3)^2*BETA(n-1, m)-(2*n-4)! od: for m from 2 to mmax do for n from m+1 to nmax do BETA(n, m) := (2*n-3)^2*BETA(n-1, m) - BETA(n-1, m-1) od: od: for n from 2 to nmax do s1(n) := 0: for m from 1 to n-1 do s1(n) := s1(n) + BETA(n, m) od: od: seq(s1(n), n=2..nmax);
# End first program
nmax := nmax; A120778 := proc(n): numer(sum(binomial(2*k1, k1)/(k1+1) / 4^k1, k1=0..n)) end proc: A000165 := proc(n): 2^n*n! end proc: A049606 := proc(n): denom(2^n/n!) end proc: for n from 2 to nmax do s2(n) := (-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) end do: seq(s2(n), n=2..nmax);
# End second program

BETA[2, 1] = -1; BETA[n_, 1] := BETA[n, 1] = (2*n - 3)^2*BETA[n - 1, 1] - (2*n - 4)!; BETA[n_ /; n > 2, m_ /; m > 0] /; 1 <= m <= n := BETA[n, m] = (2*n - 3)^2*BETA[n - 1, m] - BETA[n - 1, m - 1]; BETA[, ] = 0;
Table[Sum[BETA[n, m], {m, 1, n - 1}], {n, 2, 14}] (* Jean-François Alcover, Dec 13 2017 *)

Rowsums(n) = (-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) for n >= 2.
Conjecture: a(n) = (2*n-3)! - 2^(2*n-3)*(n-1)!*(n-2)!, for n >= 2 (gives the first 13 terms). - Christopher P. Herzog, Nov 25 2014
Meijer's and Herzog's conjectures can also be written as: a(n) = -A129890(n-2)*A000165(n-2) = A009445(n-2) - A002474(n-2). - Peter Luschny, Dec 01 2014

a(15)-a(16) from Stefano Spezia, Jun 28 2024

A061403 Denominators in the series for Bessel function J4(x).

Original entry on oeis.org

384, 7680, 368640, 30965760, 3963617280, 713451110400, 171228266496000, 52738306080768000, 20251509535014912000, 9477706462386978816000, 5307515618936708136960000, 3502960308498227370393600000

Offset: 0

Frank Ellermann, Jun 11 2001

G. C. Greubel, Table of n, a(n) for n = 0..220 (terms 0..50 from T. D. Noe)
Index entries for sequences related to Bessel functions or polynomials

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Denominator[Take[CoefficientList[Series[BesselJ[4, x], {x, 0, 500}], x], {5, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

A061404 Denominators in the series for Bessel function J5(x).

Original entry on oeis.org

3840, 92160, 5160960, 495452160, 71345111040, 14269022208000, 3767021862912000, 1265719345938432000, 526539247910387712000, 265375780946835406848000, 159225468568101244108800000

Offset: 0

Frank Ellermann, Jun 11 2001

G. C. Greubel, Table of n, a(n) for n = 0..220 (terms 0..50 from T. D. Noe)
Index entries for sequences related to Bessel functions or polynomials

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Denominator[Take[CoefficientList[Series[BesselJ[5, x], {x, 0, 500}], x], {6, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

A061405 Denominators in the series for Bessel function J6(x).

Original entry on oeis.org

46080, 1290240, 82575360, 8918138880, 1426902220800, 313918488576000, 90408524709888000, 32908702994399232000, 14743098941490855936000, 7961273428405062205440000, 5095214994179239811481600000

Offset: 0

Frank Ellermann, Jun 11 2001

G. C. Greubel, Table of n, a(n) for n = 0..220 (terms 0..50 from T. D. Noe)
Index entries for sequences related to Bessel functions or polynomials

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Denominator[Take[CoefficientList[Series[BesselJ[6, x], {x, 0, 500}], x], {7, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

A061407 Denominators in the series for Bessel function J8(x).

Original entry on oeis.org

10321920, 371589120, 29727129600, 3923981107200, 753404372582400, 195885136871424000, 65817405988798464000, 27643310515295354880000, 14153374983831221698560000, 8661865490104707679518720000

Offset: 0

Frank Ellermann, Jun 11 2001

G. C. Greubel, Table of n, a(n) for n = 0..220 (terms 0..50 from T. D. Noe)
Index entries for sequences related to Bessel functions or polynomials

J1: A002474, J2: A002506, J3: A014401 for formula etc.

Denominator[Take[CoefficientList[Series[BesselJ[8, x], {x, 0, 50}], x], {9, -1, 2}]] (* G. C. Greubel, Aug 15 2017 *)

cp's OEIS Frontend

A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

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PARI

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A002506 Denominators of coefficients of expansion of Bessel function J_2(x).

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A014401 Denominators of coefficients of expansion of Bessel function J_3(x).

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Maple

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A001900 Successive numerators of Wallis's approximation to Pi/2 (unreduced).

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A161736 Denominators of the column sums of the BG2 matrix.

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A160481 Row sums of the Beta triangle A160480.

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Maple

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A061403 Denominators in the series for Bessel function J4(x).

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