A001818 -id:A001818 - cp's OEIS frontend

A111595 Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).

1, 0, 1, 1, -2, 1, 0, 9, -6, 1, 9, -36, 42, -12, 1, 0, 225, -300, 130, -20, 1, 225, -1350, 2475, -1380, 315, -30, 1, 0, 11025, -22050, 15435, -4620, 651, -42, 1, 11025, -88200, 220500, -182280, 67830, -12600, 1204, -56, 1, 0, 893025, -2381400, 2302020, -1020600, 235494, -29736, 2052, -72

Offset: 0

Wolfdieter Lang, Aug 23 2005

This is a Sheffer triangle (lower triangular exponential convolution matrix). For Sheffer row polynomials see the S. Roman reference and explanations under A048854.
In the umbral notation of the S. Roman reference this would be called Sheffer for ((sqrt(1-2*t))/(1-t), t/(1-t)).
The associated Sheffer triangle is A111596.
Matrix logarithm equals A112239. - Paul D. Hanna, Aug 29 2005
The row polynomials (1/2^n)* H(n,sqrt(x/2))^2, with the Hermite polynomials H(n,x), have e.g.f. (1/sqrt(1-y^2))*exp(x*y/(1+y)).
The row polynomials s(n,x):=sum(a(n,m)*x^m,m=0..n), together with the associated row polynomials p(n,x) of A111596, satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
The unsigned column sequences are: A111601, A111602, A111777-A111784, for m=1..10.

			The triangle a(n, m) begins:
n\m       0         1         2          3         4         5       6       7     8    9  10 ...
0:        1
1:        0         1
2:        1        -2         1
3:        0         9        -6          1
4:        9       -36        42        -12         1
5:        0       225      -300        130       -20         1
6:      225     -1350      2475      -1380       315       -30       1
7:        0     11025    -22050      15435     -4620       651     -42       1
8:    11025    -88200    220500    -182280     67830    -12600    1204     -56     1
9:        0    893025  -2381400    2302020  -1020600    235494  -29736    2052   -72    1
10:  893025  -8930250  28279125  -30958200  15961050  -4396140  689850  -63000  3285  -90   1
-------------------------------------------------------------------------------------------------

R. P. Boas and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer, 1958, p. 41
S. Roman, The Umbral Calculus, Academic Press, New York, 1984, p. 128.

G. C. Greubel, Rows n=0..100 of triangle, flattened

Row sums: A111882. Unsigned row sums: A111883.

Cf. A112239 (matrix log).

row[n_] := CoefficientList[ 1/2^n*HermiteH[n, Sqrt[x/2]]^2, x]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 17 2013 *)

from sympy import hermite, Poly, sqrt, symbols
x = symbols('x')
def a(n): return Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()[::-1]
for n in range(11): print(a(n)) # Indranil Ghosh, May 26 2017

E.g.f. for column m>=0: (1/sqrt(1-x^2))*((x/(1+x))^m)/m!.

a(n, m)=((-1)^(n-m))*(n!/m!)*sum(binomial(2*k, k)*binomial(n-2*k-1, m-1)/(4^k), k=0..floor((n-m)/2)), n>=m>=1. a(2*k, 0)= ((2*k)!/(k!*2^k))^2 = A001818(k), a(2*k+1) = 0, k>=0. a(n, m)=0 if n

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

Original entry on oeis.org

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

A060524 Triangle read by rows: T(n,k) = number of degree-n permutations with k odd cycles, k=0..n, n >= 0.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 5, 0, 1, 9, 0, 14, 0, 1, 0, 89, 0, 30, 0, 1, 225, 0, 439, 0, 55, 0, 1, 0, 3429, 0, 1519, 0, 91, 0, 1, 11025, 0, 24940, 0, 4214, 0, 140, 0, 1, 0, 230481, 0, 122156, 0, 10038, 0, 204, 0, 1, 893025, 0, 2250621, 0, 463490, 0, 21378, 0, 285, 0, 1, 0

Offset: 0

Vladeta Jovovic, Apr 01 2001

The row polynomials t(n,x):=Sum_{k=0..n} T(n,k)*x^k satisfy the recurrence relation t(n,x) = x*t(n-1,x) + ((n-1)^2)*t(n-2,x); t(-1,x)=0, t(0,x)=1. - Wolfdieter Lang, see above.
This is an example of a Sheffer triangle (coefficient triangle for Sheffer polynomials). In the umbral calculus (see the Roman reference given under A048854) s(n,x) := Sum_{k=0..n} T(n,k)*x^k would be called Sheffer polynomials for (1/cosh(t),tanh(t)), which translates to the e.g.f. for column number k>=0 given by (1/sqrt(1-x^2))*((arctanh(x))^k)/k!. The e.g.f. given below is rewritten in this Sheffer context as (1/sqrt(1-x^2))*exp(y*log(sqrt((1+x)/(1-x))))= (1/sqrt(1-x^2))*exp(y*arctanh(x)). The rows of the Jabotinsky type triangle |A049218| provide the coefficients of the associated polynomials. - Wolfdieter Lang, Feb 24 2005
The solution of the differential-difference relation f(n+1,x)= (d/dx)f(n,x) + (n^2)*f(n-1,x), n >= 1, with inputs f(0,x) and f(1,x) = (d/dx)f(0,x) is f(n,x) = t(n,d_x)*f(0,x), with the differential operator d_x:=d/dx and the row polynomials t(n,x) defined above. This problem appears in a computation of thermo field dynamics where f(0,x)=1/cosh(x). See the triangle A060081. - Wolfdieter Lang, Feb 24 2005
The inverse of the Sheffer matrix T with elements T(n,k) is the Sheffer matrix A060081. - Wolfdieter Lang, Jul 22 2005
T(n,k)=0 if n-k= 1(mod 2), else T(n,k) = sum of M2(n,p), p from {1,...,A000041(n)} restricted to partitions with exactly k odd parts and any nonnegative number of even parts. For the M2-multinomial numbers in A-St order see A036039(n,p). - Wolfdieter Lang, Aug 07 2007

			Triangle begins:
  [1],
  [0, 1],
  [1, 0, 1],
  [0, 5, 0, 1],
  [9, 0, 14, 0, 1],
  [0, 89, 0, 30, 0, 1],
  [225, 0, 439, 0, 55, 0, 1],
  [0, 3429, 0, 1519, 0, 91, 0, 1],
  [11025, 0, 24940, 0, 4214, 0, 140, 0, 1],
  [0, 230481, 0, 122156, 0, 10038, 0, 204, 0, 1],
  [893025, 0, 2250621, 0, 463490, 0, 21378, 0, 285, 0, 1],
  [0, 23941125, 0, 14466221, 0, 1467290, 0, 41778, 0, 385, 0, 1],
  ...
Signed version begins:
  [1],
  [0, 1],
  [-1, 0, 1],
  [0, -5, 0, 1],
  [9, 0, -14, 0, 1],
  [0, 89, 0, -30, 0, 1],
  [-225, 0, 439, 0, -55, 0, 1],
  [0, -3429, 0, 1519, 0, -91, 0, 1],
  ...
From _Peter Bala_, Feb 23 2024: (Start)
Maple can verify the following series for Pi:
Row 1 polynomial R(1, x) = x:
Pi = 3 + 4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(1, 2*n)*R(1, 2*n+2)).
Row 3 polynomial R(3, x) = 5*x + x^3:
(3/2)^2 * Pi = 7 + 4*(3^4)*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(3, 2*n)*R(3, 2*n+2)).
Row 5 polynomial R(5, x) = 89*x + 30*x^3 + x^5:
((3*5)/(2*4))^2 * Pi = 11 + 4*(3*5)^4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(5, 2*n)*R(5, 2*n+2)). (End)

Alois P. Heinz, Rows n = 0..140, flattened
Marcelo Aguiar, Sarah Brauner, and Vic Reiner, Configuration spaces and peak representations, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 13. See p. 12.
Peter Bala, Meixner polynomials and Brouncker's continued fraction for Pi
Paul L. Butzer and Tom H. Koornwinder, Josef Meixner: His life and his orthogonal polynomials, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
Adel Hamdi and Jiang Zeng, Orthogonal polynomials and operator orderings, J. Math. Phys., 51:043506, (2010); arXiv:1006.0808 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Meixner polynomial of the first kind.

Cf. A060338, A060523, A094368, A028353 (col 1), A103916 (col 2), A103917 (col 3), A103918 (col 4).

Cf. A111594 (associated Sheffer polynomials), A142979, A142983.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1)*
      `if`(irem(i, 2)=1, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 09 2015
# alternative
A060524 := proc(n,k)
    option remember;
    if nR. J. Mathar, Jul 06 2023

nn = 6; Range[0, nn]! CoefficientList[
   Series[(1 - x^2)^(-1/2) ((1 + x)/(1 - x))^(y/2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 28 2012 *)

E.g.f.: (1+x)^((y-1)/2)/(1-x)^((y+1)/2).
T(n, k) = T(n-1, k-1) + ((n-1)^2)*T(n-2, k); T(-1, k):=0, T(n, -1):=0, T(0, 0)=1, T(n, k)=0 if nWolfdieter Lang, see above.
The Meixner polynomials defined by S_0(x)=1, S_1(x) = x; S_{n+1}(x) = x*S_n(x) - n^2*S_{n-1}(x) give a signed version of this triangle (cf. A060338). - N. J. A. Sloane, May 30 2013
From Peter Bala, Apr 10 2024: (Start)
The n-th row polynomial R(n, x) satisfies
(4*n + 2)*R(n, x) = (x + 1)*R(n, x+2) - (x - 1)*R(n, x-2).
Series for Pi involving the row polynomials R(n, x): for n >= 0 there holds
((2*n + 1)!!/(2^n*n!))^2 * Pi = (4*n + 3) + 4*((2*n + 1)!!^4) * Sum_{k >= 1} (-1)^(k+1)/((2*k + 1)*R(2*n+1, 2*k)*R(2*n+1, 2*k+2)). Cf. A142979 and A142983.
R(2*n, 0) = A001147(n)^2 = A001818(n); R(2*n+1, 0) = 0.
R(n, 1) = n! = A000142(n).
R(2*n, 2) = (4*n + 1)*A001147(n)^2 = (4*n + 1)*((2*n)!/(2^n*n!))^2;
R(2*n+1, 2) = 2*A001447(n+1)^2 = 2*(2*n + 1)!^2/(n!^2*4^n).
R(n, 3) = (2*n + 1)*n! = A007680(n). (End)

A177145 Expansion of e.g.f. arcsin(x).

Original entry on oeis.org

1, 0, 1, 0, 9, 0, 225, 0, 11025, 0, 893025, 0, 108056025, 0, 18261468225, 0, 4108830350625, 0, 1187451971330625, 0, 428670161650355625, 0, 189043541287806830625, 0, 100004033341249813400625, 0, 62502520838281133375390625, 0, 45564337691106946230659765625, 0

Offset: 1

Michel Lagneau, May 03 2010

A001818 interspersed with zeros. - Joerg Arndt, Aug 31 2013
a(n) is the number of permutations of n-1 where all cycles have even length.  For example, a(5)=9 and the permutations of 4 elements with only even cycles are (1,2)(3,4); (1,3)(2,4); (1,4)(2,3); (1,2,3,4); (1,2,4,3); (1,3,2,4); (1,3,4,2); (1,4,2,3); (1,4,3,2).
a(n) is the number of permutations on n - 1 elements where there are no cycles of even length and an even number of cycles of odd length. - N. Sato, Aug 29 2013

			1 is in the sequence because, for k=1, f'(x) = 1/sqrt(1-x^2), and f'(0) = 1.
G.f. = x + x^3 + 9*x^5 + 225*x^7 + 11025*x^9 + 893025*x^11 + ...

L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.

Muhammad Adam Dombrowski and Gregory Dresden, Areas Between Cosines, arXiv:2404.17694 [math.CO], 2024. See p. 11.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, p. 12.

Alternate terms are A001818. - N. Sato, May 13 2010

Cf. A087137.

n0:= 30: T:=array(1..n0+1): f:=x->arcsin(x):for n from 1 to n0 do:T[n]:=(D(f)(0)):f:=D(f):od: print(T):

a[ n_] := If[ n < 1, 0, If[ EvenQ[n], 0, (n - 2)!!^2]]; (* Michael Somos, Oct 07 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ ArcSin[x], {x, 0, n}]]; (* Michael Somos, Oct 07 2013 *)

Vec( serlaplace( sqrt( 1/(1-x^2) + O(x^55) ) ) )

{a(n) = if( n<2, n==1, (n-2)^2 * a(n-2))}; /* Michael Somos, Oct 07 2013 */

a(n) = if( n<0, 0, n! * polcoeff( asin(x + x * O(x^n)), n)); /* Michael Somos, Oct 07 2013 */

E.g.f.: arcsin(x).
G.f.: Q(0)*x^2/(1+x) + x/(1+x), where Q(k) = 1 + (2*k + 1)^2 * x * (1 + x * Q(k+1));  - Sergei N. Gladkovskii, May 10 2013 [Edited by Michael Somos, Oct 07 2013]
E.g.f of a(n+1), n >= 0, is 1/sqrt(1 - x^2). - N. Sato, Aug 29 2013
If n is odd, a(n) ~ 2*n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 05 2013
E.g.f.: arcsin(x) = x + x^3/(T(0)-x^2), where T(k) = 4*k^2*(1+x^2) + 2*k*(5+2*x^2) +6 + x^2 - 2*x^2*(k+1)*(2*k+3)^3/T(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 13 2013
a(n) = (n-1)! - A087137(n-1). - Anton Zakharov, Oct 18 2016
From Peter Bala, Aug 09 2024: (Start)
a(2*n+1) = (2*n - 1)!!^2 = A001147(n)^2.
a(n) = (n - 2)^2 * a(n-2) with a(1) = 1 and a(2) = 0. (End)

A060063 Triangle of coefficients of certain polynomials used for G.f.s of columns of triangle A060058.

Original entry on oeis.org

1, 1, 1, 5, 26, 9, 61, 775, 1179, 225, 1385, 32516, 114318, 87156, 11025, 50521, 1894429, 11982834, 20371266, 9652725, 893025, 2702765, 148008446, 1472351967, 4417978068, 4546174779, 1502513550

Offset: 0

Wolfdieter Lang, Mar 16 2001

The row polynomials p(n,x) (rising powers of x) appear as numerators of the column g.f.s of triangle A060058.
First column (m=0) gives A000364 (Euler numbers). See A091742, A091743, A091744 for columns m=1..3.
The main diagonal gives A001818. The row sums give A052502. The alternating row sums give A091745.

			Triangle begins:
  {1};
  {1,1};
  {5,26,9};     <-- p(2,n)=5+26*x+9*x^2.
  {61,775,1179,225};
  ...

W. Lang, First 8 rows.

The row polynomials p(n, x) := Sum_{m=0..n} a(n, m)*x^m satisfy the differential equation: p(n, x) = x*((1-x)^2)*(d^2/dx^2)p(n-1, x) + (1+6*(n-1)*x+(5-6*n)*x^2)*(d/dx)p(n-1, x) + (3*n-2)*(1+(3*n-2)*x)*p(n-1, x), n >= 1, with input p(0, x)=1. - Wolfdieter Lang, Feb 13 2004

A081919 E.g.f.: exp(x)/sqrt(1-x^2).

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 376, 1912, 17984, 119296, 1438336, 11749376, 172665472, 1674715264, 29022277376, 325841353216, 6504163557376, 82954203410432, 1874028623417344, 26760916479840256, 674914911967133696

Offset: 0

Paul Barry, Apr 01 2003

Binomial transform of aerated A001818 = 1, 0, 1, 0, 9, 0, 225, ... .

Number of perfect matchings in graph P_{2} X K_{n}. - Andrew Howroyd, Feb 28 2016

Robert Israel, Table of n, a(n) for n = 0..448

Cf. A081920.

f:= gfun:-rectoproc({-a(n) +a(n-1) +(n-1)^2*a(n-2) -(n-1)*(n-2)*a(n-3)=0, a(0) = 1, a(1)=1,a(2)=2},a(n),remember):
map(f, [$0..25]); # Robert Israel, Feb 28 2016

CoefficientList[Series[E^x/Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 04 2014 *)
a[n_] := Sum[((2k-1)!!)^2 Binomial[n, 2k], {k, 0, n/2}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

x='x+O('x^30); Vec(serlaplace(exp(x)/sqrt(1-x^2))) \\ Michel Marcus, Aug 21 2014

A081919 = lambda n: hypergeometric([1/2,-n/2,(1-n)/2], [], 4)
[round(A081919(n).n()) for n in range(21)] # Peter Luschny, Aug 21 2014

D-finite with recurrence: -a(n) +a(n-1) +(n-1)^2*a(n-2) -(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 09 2012
a(n) ~ n^n * (exp(1)+(-1)^n*exp(-1)) / exp(n). - Vaclav Kotesovec, Feb 04 2014
a(n) = hyper3F0([1/2,-n/2,(1-n)/2],[],4). - Peter Luschny, Aug 21 2014
a(n) = sum_{k=0..floor(n/2)} ((2*k-1)!!)^2*binomial(n, 2*k). - Andrew Howroyd, Feb 28 2016
E.g.f. A(x) satisfies (1-x^2)*A'(x) - (1+x-x^2)*A(x) = 0, from which R. J. Mathar's recurrence follows. - Robert Israel, Feb 28 2016

A162443 Numerators of the BG1[ -5,n] coefficients of the BG1 matrix.

Original entry on oeis.org

5, 66, 680, 2576, 33408, 14080, 545792, 481280, 29523968, 73465856, 27525120, 856162304, 1153433600, 18798870528, 86603988992, 2080374784, 2385854332928, 3216930504704, 71829033058304, 7593502179328, 281749854617600

Offset: 1

Johannes W. Meijer, Jul 06 2009

The BG1 matrix coefficients are defined by BG1[2m-1,1] = 2*beta(2m) and the recurrence relation BG1[2m-1,n] = BG1[2m-1,n-1] - BG1[2m-3,n-1]/(2*n-3)^2 with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. . As usual beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). For the BG2 matrix, the even counterpart of the BG1 matrix, see A008956.
We discovered that the n-th term of the row coefficients can be generated with BG1[1-2*m,n] = RBS1(1-2*m,n)* 4^(n-1)*((n-1)!)^2/ (2*n-2)! for m >= 1. For the BS1(1-2*m,n) polynomials see A160485.
The coefficients in the columns of the BG1 matrix, for m >= 1 and n >= 2, can be generated with GFB(z;n) = ((-1)^(n+1)*CFN2(z;n)*GFB(z;n=1) + BETA(z;n))/((2*n-3)!!)^2 for n >= 2. For the CFN2(z;n) and the Beta polynomials see A160480.
The BG1[ -5,n] sequence can be generated with the first Maple program and the BG1[2*m-1,n] matrix coefficients can be generated with the second Maple program.
The BG1 matrix is related to the BS1 matrix, see A160480 and the formulas below.

			The first few formulas for the BG1[1-2*m,n] matrix coefficients are:
BG1[ -1,n] = (1)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -3,n] = (1-2*n)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!
The first few generating functions GFB(z;n) are:
GFB(z;2) = ((-1)*(z^2-1)*GFB(z;1) + (-1))/1
GFB(z;3) = ((+1)*(z^4-10*z^2+9)*GFB(z;1) + (-11 + z^2))/9
GFB(z;4) = ((-1)*( z^6- 35*z^4+259*z^2-225)*GFB(z;1) + (-299 + 36*z^2 - z^4))/225

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
J. M. Amigo, Relations among Sums of Reciprocal Powers Part II, International Journal of Mathematics and Mathematical Sciences , Volume 2008 (2008), pp. 1-20.

A162444 are the denominators of the BG1[ -5, n] matrix coefficients.
The BG1[ -3, n] equal (-1)*A002595(n-1)/A055786(n-1) for n >= 1.
The BG1[ -1, n] equal A046161(n-1)/A001790(n-1) for n >= 1.
The cs(n) equal A046161(n-2)/A001803(n-2) for n >= 2.
The BETA(z, n) polynomials and the BS1 matrix lead to the Beta triangle A160480.
The CFN2(z, n), the t2(n, m) and the BG2 matrix lead to A008956.
Cf. A000364, A001818 and A160485.
Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).

a := proc(n): numer((1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!) end proc: seq(a(n), n=1..21);
# End program 1
nmax1 := 5; coln := 3; Digits := 20: mmax1 := nmax1: for n from 0 to nmax1 do t2(n, 0) := 1 od: for n from 0 to nmax1 do t2(n, n) := doublefactorial(2*n-1)^2 od: for n from 1 to nmax1 do for m from 1 to n-1 do t2(n, m) := (2*n-1)^2* t2(n-1, m-1) + t2(n-1, m) od: od: for m from 1 to mmax1 do BG1[1-2*m, 1] := euler(2*m-2) od: for m from 1 to mmax1 do BG1[2*m-1, 1] := Re(evalf(2*sum((-1)^k1/(1+2*k1)^(2*m), k1=0..infinity))) od: for m from -mmax1 +coln to mmax1 do BG1[2*m-1, coln] := (-1)^(coln+1)*sum((-1)^k1*t2(coln-1, k1)*BG1[2*m-(2*coln-1)+2*k1, 1], k1=0..coln-1)/doublefactorial(2*coln-3)^2 od;
# End program 2
# Maple programs edited by Johannes W. Meijer, Sep 25 2012

a(n) = numer(BG1[ -5,n]) and A162444(n) = denom(BG1[ -5,n]) with BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!.
The generating functions GFB(z;n) of the coefficients in the matrix columns are defined by
GFB(z;n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
GFB(z;n) = (1-z^2/(2*n-3)^2)*GFB(n-1) - 4^(n-2)*(n-2)!^2/((2*n-4)!*(2*n-3)^2) for n => 2 with GFB(z;n=1) = 1/(z*cos(Pi*z/2))*int(sin(z*t)/sin(t),t=0..Pi/2).
The column sums cs(n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity) = 4^(n-1)/((2*n-2)*binomial(2*n-2,n-1)) for n >= 2.
BG1[2*m-1,n] = (n-1)!^2*4^(n-1)*BS1[2*m-1,n]/(2*n-2)!

A296675 Expansion of e.g.f. 1/(1 - arcsinh(x)).

Original entry on oeis.org

1, 1, 2, 5, 16, 69, 368, 2169, 14208, 109929, 970752, 8995821, 88341504, 988161069, 12276025344, 154843019169, 2009594658816, 29484826539345, 476778061430784, 7588488203093205, 121001549512310784, 2205431202369899925, 44538441694414110720, 852615914764223422665

Offset: 0

Ilya Gutkovskiy, Dec 18 2017

sign

a(48) is negative. - Vaclav Kotesovec, Jan 26 2020

			1/(1 - arcsinh(x)) = 1 + x/1! + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 69*x^5/5! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A000111, A001818, A006154, A189780, A385343.

a:=series(1/(1-arcsinh(x)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[1/(1 - ArcSinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[1/(1 - Log[x + Sqrt[1 + x^2]]), {x, 0, nmax}], x] Range[0, nmax]!

x='x+O('x^99); Vec(serlaplace(1/(1-log(x+sqrt(1+x^2))))) \\ Altug Alkan, Dec 18 2017

E.g.f.: 1/(1 - log(x + sqrt(1 + x^2))).
a(n) ~ 8*((4 - Pi^2)*sin(Pi*n/2) - 4*Pi*cos(Pi*n/2)) * n^(n-1) / ((4 + Pi^2)^2 * exp(n)). - Vaclav Kotesovec, Dec 18 2017
a(n) = Sum_{k=0..n} k! * i^(n-k) * A385343(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 27 2025

A092396 Row 2 of array in A288580.

Original entry on oeis.org

1, -1, -4, 9, 64, -225, -2304, 11025, 147456, -893025, -14745600, 108056025, 2123366400, -18261468225, -416179814400, 4108830350625, 106542032486400, -1187451971330625, -34519618525593600, 428670161650355625, 13807847410237440000, -189043541287806830625, -6682998146554920960000

Offset: 0

J. Dezert (Jean.Dezert(AT)onera.fr), Mar 21 2004

sign

			!3!_2 = 3(3-2)(3-4)(3-6) = 9 and !4!_2 = 4(4-2)(4-6)(4-8) = 64.

F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.

Cf. A288580, A001818.

T:=proc(n,k)  local i,p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
r:=k->[seq(T(n,k), n=0..60)]; r(2); # N. J. A. Sloane, Jul 03 2017

T[n_, k_] := Module[{i, p = 1}, For[i = 0, i <= Floor[2n/k], i++, If[n - k i != 0, p *= (n - k i)]]; p]; T[_, 0] = 1;
Table[T[n, 2], {n, 0, 22}] (* Jean-François Alcover, Apr 05 2020, after Maple *)

a(n) = !n!2 = Prod{i=0, 1, 2, ... .}_{0<|n-2i|<=n}(n-2i) = n(n-2)(n-4)... = (-1)^(Int(n+1)/2) * (n!!)^2

Entry revised by N. J. A. Sloane, Jul 03 2017

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

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A111595 Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).

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

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A060524 Triangle read by rows: T(n,k) = number of degree-n permutations with k odd cycles, k=0..n, n >= 0.

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A177145 Expansion of e.g.f. arcsin(x).

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A060063 Triangle of coefficients of certain polynomials used for G.f.s of columns of triangle A060058.

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A081919 E.g.f.: exp(x)/sqrt(1-x^2).

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A162443 Numerators of the BG1[ -5,n] coefficients of the BG1 matrix.

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