A002866 -id:A002866 - cp's OEIS frontend

A144828 Partial products of successive terms of A017113; a(0)=1.

1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400, 9033331507200, 686533194547200, 57668788341964800, 5305528527460761600, 530552852746076160000, 57299708096576225280000, 6646766139202842132480000, 824199001261152424427520000, 108794268166472120024432640000

Offset: 0

Philippe Deléham, Sep 21 2008

a(n) is the number of signed permutations of length 4n that are equal to their reverse-inverses. Note that the reverse-inverse of a permutation is equivalent to a 90-degree rotation of the permutation's diagram (see the Hardt and Troyka link). - Justin M. Troyka, Aug 11 2011

Define the bar operation as an operation on signed permutation that flips the sign of each entry. Then a(n) is the number of signed permutations of length 2n that are equal to the bar of their inverses and equal to their reverse-complements (see the Hardt and Troyka link). - Justin M. Troyka, Aug 11 2011

			a(0)=1, a(1)=4, a(2)=4*12=48, a(3)=4*12*20=960, a(4)=4*12*20*28=26880, ...
Since a(1) = 4, there are 4 signed permutations of 4 that are equal to their reverse-inverses.  These are: (+2,+4,+1,+3), (+3,+1,+4,+2), (-2,-4,-1,-3), (-3,-1,-4,-2). - _Justin M. Troyka_, Aug 11 2011
G.f. = 1 + 4*x + 48*x^2 + 960*x^3 + 26880*x^4 + 967680*x^5 + 42577920*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179-217.
A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).

Cf. A000108, A001715, A002866, A007559, A008546, A017113, A047053, A048994, A049308, A144827.
Essentially the same as A052714. - N. J. A. Sloane, Feb 03 2013
Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or this sequence) (m=2), A221954 (m=3), A052734 (m=4), A221953 (m=5), A221955 (m=6).

[2^k *Factorial(2*k) / Factorial(k): k in [0..20]]; // Vincenzo Librandi, Aug 11 2011

A144828:= n-> 2^n*n!*binomial(2*n,n); seq(A144828(n), n=0..30); # G. C. Greubel, Apr 02 2021

Table[4^n (2 n - 1)!!, {n, 0, 15}] (* Vincenzo Librandi, May 14 2015 *)
Join[{1},FoldList[Times,(8*Range[0,20]+4)]] (* Harvey P. Dale, Dec 01 2015 *)

a(n)=binomial(2*n,n)*n!<Charles R Greathouse IV, Jan 17 2012

{a(n) = if( n<0, (-1)^n / a(-n), 2^n *(2*n)! / n!)}; /* Michael Somos, Jan 06 2017 */

[2^n*factorial(n+1)*catalan_number(n) for n in (0..30)] # G. C. Greubel, Apr 02 2021

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*8^(n-k).
a(n) = A052714(n+1). - R. J. Mathar, Oct 01 2008
a(n) = 2^n *(2*n)! / n!. - Justin M. Troyka, Aug 11 2011
G.f.: 1/(1-4x/(1-8x/(1-12x/(1-16x/(1-20x/(1-24x/(1-28x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (-4)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
E.g.f.: 1/sqrt(1-8*x). - Philippe Deléham, May 14 2015
a(n) = 4^n * A001147(n). - Philippe Deléham, May 14 2015
a(n) = 8^n * Gamma(n + 1/2) / sqrt(Pi). - Daniel Suteu, Jan 06 2017
0 = a(n)*(8*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) and a(n) = (-1)^n / a(-n) for all n in Z. - Michael Somos, Jan 06 2017
a(n) = 2^n * (n+1)! * Catalan(n). - G. C. Greubel, Apr 02 2021
Sum_{n>=0} 1/a(n) = 1 + e^(1/8)*sqrt(Pi)*erf(1/(2*sqrt(2)))/(2*sqrt(2)), where erf is the error function. - Amiram Eldar, Dec 20 2022

A167584 The ED4 array read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 13, 6, 1, 76, 41, 10, 1, 789, 372, 93, 14, 1, 7734, 4077, 1020, 169, 18, 1, 110937, 53106, 13269, 2212, 269, 22, 1, 1528920, 795645, 198990, 33165, 4140, 393, 26, 1, 28018665, 13536360, 3383145, 563850, 70485, 6996, 541, 30, 1

Offset: 1

Johannes W. Meijer, Nov 10 2009

The coefficients in the upper right triangle of the ED4 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED4 array (m<=n) were found with the recurrence relation, see below. We use for the array rows the letter n (>=1) and for the array columns the letter m (>=1).
For the ED1, ED2 and ED3 arrays see A167546, A167560 and A167572.
The Madhava-Gregory-Leibniz series representation for Pi/4 is the case m = 0 of the following more general result: for m = 0,1,2,... there holds 1/(2*m)! * Pi/4 = Sum_{k >= 0} ( (-1)^(m+k) * 1/Product_{j = -m .. m} (2*k + 1 + 2*j) ). The entries of this table are given by truncating these series to n-1 terms and then scaling by certain double factorials -- see the formula below. - Peter Bala, Nov 06 2016

			The ED4 array begins with:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1
  2, 6, 10, 14, 18, 22, 26, 30, 34, 38
  13, 41, 93, 169, 269, 393, 541, 713, 909, 1129
  76, 372, 1020, 2212, 4140, 6996, 10972, 16260, 23052, 31540
  789, 4077, 13269, 33165, 70485, 133869, 233877, 382989, 595605, 888045
  7734, 53106, 198990, 563850, 1339110, 2812194, 5389566, 9619770, 16216470, 26081490
  ...
From _Peter Bala_, Nov 06 2016: (Start)
Table extended to nonpositive values of m:
  n\m|     -4     -3    -2    -1    0
  -----------------------------------
   0 |      0      0     0     0    0
   1 |      1      1     1     1    1
   2 |    -18    -14   -10    -6   -2
   3 |    233    141    73    29    9
   4 |  -2844  -1428  -620  -228  -60
   5 |  39309  17877  7149  2325  525
  ...
Column  0: (-1)^(n+1)*(2*n - 3)!!*n. See A001193;
Column -1: (-1)^n*(2*n - 5)!!/3!!*n*(7 - 4*n^2);
Column -2: (-1)^n*(2*n - 7)!!/5!!*n(-149 + 120*n^2 - 16*n^4);
Column -3: (-1)^n*(2*n - 9)!!/7!!*n*(6483 - 6076*n^2 + 1232*n^4 - 64*n^6);
Column -4: (-1)^n*(2*n - 11)!!/9!!*n*(-477801 + 489136*n^2 - 120288*n^4 + 9984*n^6 - 256*n^8). (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johannes W. Meijer, The four Escher-Droste arrays, jpg image, Mar 08 2013.
Wikipedia, Double factorial

A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the array.
A024199, A167588 and A167589 equal the first three columns of the array.
A167590 equals the row sums of the ED4 array read by antidiagonals.
A167591 is a triangle related to the a(n) formulas of the rows of the ED4 array.
A167594 is a triangle related to the GF(z) formulas of the rows of the ED4 array.
Cf. A002866 (the 2^(n-1)*n! factor).
Cf. A167546 (ED1 array), A167560 (ED2 array), A167572 (ED3 array). Cf. A001193, A003881.

T := proc (n, m) option remember;
      if n = 0 then 0
       elif n = 1 then 1
       else (4*m-2)*T(n-1,m)+(2*n+2*m-5)*(2*n-2*m-1)*T(n-2,m)
      end if;
     end proc:
#square array read by antidiagonals
seq(seq(T(n-m,m), m = 1..n-1), n = 1..10);
# Peter Bala, Nov 06 2016

T[0, k_] := 0; T[1, k_] := 1; T[n_, k_] := T[n, k] = (4*k - 2)*T[n - 1, k] + (2*n + 2*k - 5)*(2*n - 2*k - 1)*T[n - 2, k]; Table[T[n - k, k], {n, 2, 12}, {k, 1, n - 1}] (* G. C. Greubel, Jan 20 2017 *)

a(n,m) = ((2*m-3)!!/(2*(2*m-2*n-3)!!))*Integral_{y=0..oo} sinh(y*(2*n))/(cosh(y))^(2*m-1) dy for m>n.
The (n-1)-differences of the n-th array row lead to the recurrence relation
Sum_{k=0..n-1} (-1)^k*binomial(n-1,k)*a(n,m-k) = 2^(n-1)*n!
From Peter Bala, Nov 06 2016: (Start)
T(n,m) = ((2*m - 3)!!/(2*(2*m - 2*n - 3)!!)) * Sum_{k = 0..n-1} (-1)^(k+1)*binomial(2*n - k - 1, k)*2^(2*n - 2*k - 1)*1/(2*n - 2*m - 2*k + 1), for n and m >= 0.
Note the double factorial for a negative odd integer N is defined in terms of the gamma function as N!! = 2^((N+1)/2)*Gamma(N/2 + 1)/sqrt(Pi).
T(n, m) = (2*m - 3)!! * (2*n + 2*m - 3)!! * Sum_{k = 0..n-1} ( (-1)^(m + k + 1) / Product_{j = -(m-1) .. m-1} (2*k + 1 + 2*j) ).
Using this result we can extend the table to nonpositive values of m (the column index). Column 0 is a signed version of A001193. We have for m <= 0, T(n,m) = (2*n - 2*|m| - 3)!!/(2*|m| + 1)!! * Sum_{k = 0..n-1} (-1)^k*Product_{j = -|m|..|m|} (2*k + 1 + 2*j).
Recurrence: T(n, m) = (4*m - 2)*T(n-1, m) + (2*n + 2*m - 5)*(2*n - 2*m - 1)*T(n-2, m).
For a fixed value of n, the entries in row n are polynomial in the value of the column index m. The first few polynomials are [1, 4*m - 2, 12*m^2 - 8*m + 9, 32*m^3 - 16*m^2 + 120*m - 60, 80*m^4 + 952*m^2 - 768*m + 525, ...]. (End)

A257609 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 2*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 16, 4, 8, 88, 88, 8, 16, 416, 1056, 416, 16, 32, 1824, 9664, 9664, 1824, 32, 64, 7680, 76224, 154624, 76224, 7680, 64, 128, 31616, 549504, 1999232, 1999232, 549504, 31616, 128, 256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256

Offset: 0

Dale Gerdemann, May 03 2015

			Triangle begins as:
    1;
    2,      2;
    4,     16,       4;
    8,     88,      88,        8;
   16,    416,    1056,      416,       16;
   32,   1824,    9664,     9664,     1824,       32;
   64,   7680,   76224,   154624,    76224,     7680,      64;
  128,  31616,  549504,  1999232,  1999232,   549504,   31616,    128;
  256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A002866 (row sums), A008292, A060187, A100575, A257611, A257613.
Cf. A257615, A038208, A256890, A257610, A257612, A257614, A257616, A257617.
Cf. A257618, A257619.
Similar sequences listed in A256890.

function T(n,k,a,b)
  if k lt 0 or k gt n then return 0;
  elif k eq 0 or k eq n then return 1;
  else return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b);
  end if; return T;
end function;
[T(n,k,2,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 21 2022

A257609:= func< n,k | 2^n*EulerianNumber(n+1, k) >;
[A257609(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 17 2025

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,2,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)

def T(n,k,a,b): # A257609
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,2,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.
Sum_{k=0..n} T(n, k) = A002866(n).
From G. C. Greubel, Mar 21 2022: (Start)
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 2, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = 2*A100575(n+1). (End)
T(n, k) = 2^n*A008292(n+1, k+1). - G. C. Greubel, Jan 17 2025

A051578 a(n) = (2*n+4)!!/4!!, related to A000165 (even double factorials).

Original entry on oeis.org

1, 6, 48, 480, 5760, 80640, 1290240, 23224320, 464486400, 10218700800, 245248819200, 6376469299200, 178541140377600, 5356234211328000, 171399494762496000, 5827582821924864000, 209792981589295104000, 7972133300393213952000, 318885332015728558080000

Offset: 0

Wolfdieter Lang

Row m=4 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 521.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A051577 (rows m=0..3), A051579, A051580, A051581, A051582, A051583.

Cf. A052587 (essentially the same).

List([0..20], n-> 2^(n-1)*Factorial(n+2) ); # G. C. Greubel, Nov 11 2019

[2^(n-1)*Factorial(n+2): n in [0..20]]; // G. C. Greubel, Nov 11 2019

a:= proc(n) option remember; `if`(n=0, 1, 2*(n+2)*a(n-1)) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 29 2019
seq(2^(n-1)*(n+2)!, n=0..20); # G. C. Greubel, Nov 11 2019

Table[2^(n-1)(n+2)!, {n,0,20}] (* Jean-François Alcover, Oct 05 2019 *)
Table[(2n+4)!!/8,{n,0,20}] (* Harvey P. Dale, Apr 06 2023 *)

vector(21, n, 2^(n-2)*(n+1)! ) \\ G. C. Greubel, Nov 11 2019

apply( {A051578(n)=(n+2)!<<(n-1)}, [0..18]) \\ M. F. Hasler, Nov 10 2024

[2^(n-1)*factorial(n+2) for n in (0..20)] # G. C. Greubel, Nov 11 2019

a(n) = (2*n+4)!!/4!!.
E.g.f.: 1/(1-2*x)^3.
a(n) ~ 2^(-1/2)*Pi^(1/2)*n^(5/2)*2^n*e^-n*n^n*{1 + 37/12*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
a(n) = (n+2)!*2^(n-1). - Zerinvary Lajos, Sep 23 2006. [corrected by Gary Detlefs, Apr 29 2019]
a(n) = 2^n*A001710(n+2). - R. J. Mathar, Feb 22 2008
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 6)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 6*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 6*x/(1 - 2*x/(1 - 8*x/(1 - 4*x/(1 - 10*x/(1 - 6*x/(1 - ... - (2*n + 4)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 6*x/(1 - 8*x/(1 - 2*x/(1 - 10*x/(1 - 4*x/(1 - 12*x/(1 - 6*x/(1 - ... - (2*n + 6)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 8*sqrt(e) - 12.
Sum_{n>=0} (-1)^n/a(n) = 8/sqrt(e) - 4. (End)
a(n) = A052587(n+2) for n > 0. - M. F. Hasler, Nov 10 2024

A144827 Partial products of successive terms of A017029; a(0)=1.

Original entry on oeis.org

1, 4, 44, 792, 19800, 633600, 24710400, 1136678400, 60243955200, 3614637312000, 242180699904000, 17921371792896000, 1451631115224576000, 127743538139762688000, 12135636123277455360000, 1237834884574300446720000, 134924002418598748692480000, 15651184280557454848327680000

Offset: 0

Philippe Deléham, Sep 21 2008

nonn

			a(0)=1, a(1)=4, a(2)=4*11=44, a(3)=4*11*18=792, a(4)=4*11*18*25=19800, ...

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A001715, A002866, A007559, A008546, A045754, A047053, A132393.

Cf. A049209, A049308, A051188, A084947, A144739, A147585.

[ 1 ] cat [ &*[ (7*k+4): k in [0..n] ]: n in [0..14] ]; // Klaus Brockhaus, Nov 10 2008

FoldList[Times,1,Range[4,150,7]] (* Harvey P. Dale, Apr 25 2014 *)

[1]+[4*7^(n-1)*rising_factorial(11/7, n-1) for n in (1..30)] # G. C. Greubel, Feb 22 2022

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*7^(n-k).
G.f.: 1/(1-4*x/(1-7*x/(1-11*x/(1-14*x/(1-18*x/(1-21*x/(1-25*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-3)^n*Sum_{k=0..n} (7/3)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 7*x)^(4/7).
a(n) ~ sqrt(2*Pi)*7^n*n^(n+1/14)/(exp(n)*Gamma(4/7)). (End)
a(n) = 4*7^(n-1)*Pochhammer(n-1, 11/7) with a(0) = 1. - G. C. Greubel, Feb 22 2022
Sum_{n>=0} 1/a(n) = 1 + (e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). - Amiram Eldar, Dec 19 2022

Corrected a(9) by Vincenzo Librandi, Jul 14 2011

A051580 a(n) = (2*n+6)!!/6!!, related to A000165 (even double factorials).

Original entry on oeis.org

1, 8, 80, 960, 13440, 215040, 3870720, 77414400, 1703116800, 40874803200, 1062744883200, 29756856729600, 892705701888000, 28566582460416000, 971263803654144000, 34965496931549184000, 1328688883398868992000

Offset: 0

Wolfdieter Lang

Row m=6 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A051577, A051578, A051579 (rows m=0..5), A051581, A051582, A051583.

List([0..20], n-> Product([1..n], j-> 2*j+6) ); # G. C. Greubel, Nov 11 2019

[1] cat [(&*[2*j+6: j in [1..n]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(2*j+6, j=1..n), n=0..20); # G. C. Greubel, Nov 11 2019

Table[2^n*Pochhammer[4, n], {n,0,20}] (* G. C. Greubel, Nov 11 2019 *)
Table[(2n+6)!!/6!!,{n,0,20}] (* Harvey P. Dale, Mar 03 2022 *)

vector(20, n, prod(j=1,n-1, 2*j+6) ) \\ G. C. Greubel, Nov 11 2019

[product( (2*j+6) for j in (1..n)) for n in (0..20)] # G. C. Greubel, Nov 11 2019

a(n) = (2*n+6)!!/6!!.
E.g.f.: 1/(1-2*x)^4.
a(n) = n!*2^(n-4)/3, n>=3. - Zerinvary Lajos, Sep 23 2006
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+8)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 8)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 8*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 8*x/(1 - 2*x/(1 - 10*x/(1 - 4*x/(1 - 12*x/(1 - 6*x/(1 - ... - (2*n + 6)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 8*x/(1 - 10*x/(1 - 2*x/(1 - 12*x/(1 - 4*x/(1 - 14*x/(1 - 6*x/(1 - ... - (2*n + 8)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 48*sqrt(e) - 78.
Sum_{n>=0} (-1)^n/a(n) = 30 - 48/sqrt(e). (End)

A051579 a(n) = (2*n+5)!!/5!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 7, 63, 693, 9009, 135135, 2297295, 43648605, 916620705, 21082276215, 527056905375, 14230536445125, 412685556908625, 12793252264167375, 422177324717523375, 14776206365113318125, 546719635509192770625

Offset: 0

Wolfdieter Lang

Row m=5 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A051577, A051578 (rows m=0..4).

Cf. A051580, A051581, A051582, A051583, A178647.

List([0..20], n-> Product([0..n-1], j-> 2*j+7) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+7: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

df:=doublefactorial; seq(df(2*n+5)/df(5), n = 0..20); # G. C. Greubel, Nov 12 2019

Table[2^n*Pochhammer[7/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=1,n-1, 2*j+5) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+7) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+5)!!/4!!.
E.g.f.: 1/(1-2*x)^(7/2).
a(n) ~ 8/15*sqrt(2)*n^3*2^n*e^-n*n^n*(1 + 107/24*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
G.f.: G(0)/(10*x) -1/(5*x), where G(k)= 1 + 1/(1 - x*(2*k+5)/(x*(2*k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 13 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 7)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 7*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 7*x/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - ... - (2*n + 5)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 7*x/(1 - 9*x/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - ... - (2*n + 7)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 15 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 20, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 15 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 10, where erfi is the imaginary error function. (End)

A051582 a(n) = (2*n+8)!!/8!!, related to A000165 (even double factorials).

Original entry on oeis.org

1, 10, 120, 1680, 26880, 483840, 9676800, 212889600, 5109350400, 132843110400, 3719607091200, 111588212736000, 3570822807552000, 121407975456768000, 4370687116443648000, 166086110424858624000, 6643444416994344960000

Offset: 0

Wolfdieter Lang

Row m=8 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..399
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1).

Cf. A051577, A051578, A051579, A051580, A051581 (rows m=0..7), A051583.

F:=Factorial;; List([0..20], n-> 2^n*F(n+4)/F(4) ); # G. C. Greubel, Nov 12 2019

F:=Factorial; [2^n*F(n+4)/F(4): n in [0..20]]; // G. C. Greubel, Nov 12 2019

seq(2^n*pochhammer(5, n), n=0..20); # G. C. Greubel, Nov 12 2019

(2Range[0,20]+8)!!/8!! (* Harvey P. Dale, Feb 03 2013 *)
Table[2^n*Pochhammer[5, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, n--; (n+4)!*2^(n-1)/12) \\ Michel Marcus, Feb 09 2015

f=factorial; [2^n*f(n+4)/f(4) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+8)!!/8!!.
E.g.f.: 1/(1-2*x)^5.
a(n) = (n+4)!*2^(n-1)/12. - Zerinvary Lajos, Sep 23 2006
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 10)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 10*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 10*x/(1 - 2*x/(1 - 12*x/(1 - 4*x/(1 - 14*x/(1 - 6*x/(1 - ... - (2*n + 8)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 10*x/(1 - 12*x/(1 - 2*x/(1 - 14*x/(1 - 4*x/(1 - 16*x/(1 - 6*x/(1 - ... - (2*n + 10)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 384*sqrt(e) - 632.
Sum_{n>=0} (-1)^n/a(n) = 384/sqrt(e) - 232. (End)

A324513 Number of aperiodic cycle necklaces with n vertices.

Original entry on oeis.org

1, 0, 0, 0, 2, 7, 51, 300, 2238, 18028, 164945, 1662067, 18423138, 222380433, 2905942904, 40864642560, 615376173176, 9880203467184, 168483518571789, 3041127459127222, 57926238289894992, 1161157775616335125, 24434798429947993043, 538583682037962702384

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

We define an aperiodic cycle necklace to be an equivalence class of (labeled, undirected) Hamiltonian cycles under rotation of the vertices such that all n of these rotations are distinct.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Inequivalent representatives of the a(6) = 7 aperiodic cycle necklaces.
Gus Wiseman, Inequivalent representatives of the a(7) = 51 aperiodic cycle necklaces.

Cf. A000740, A000939, A001037 (binary Lyndon words), A008965, A059966 (Lyndon compositions), A060223 (normal Lyndon words), A061417, A064852 (if cycle is oriented), A086675, A192332, A275527, A323866 (aperiodic toroidal arrays), A323871.

Cf. A306669, A324461, A324462, A324512, A324514.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#,2,1,1]]&/@Permutations[Range[n]]],#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&&UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,8}]

a(n)={if(n<3, n==0||n==1, (if(n%2, 0, -(n/2-1)!*2^(n/2-2)) + sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2))/2)} \\ Andrew Howroyd, Aug 19 2019

a(n) = A324512(n)/n.

a(2*n+1) = A064852(2*n+1)/2 for n > 0; a(2*n) = (A064852(2*n) - A002866(n-1))/2 for n > 1. - Andrew Howroyd, Aug 16 2019

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

A051581 a(n) = (2*n+7)!!/7!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 9, 99, 1287, 19305, 328185, 6235515, 130945815, 3011753745, 75293843625, 2032933777875, 58955079558375, 1827607466309625, 60311046388217625, 2110886623587616875, 78102805072741824375, 3046009397836931150625

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1).

Cf. A051577, A051578, A051579, A051580 (rows m=0..6), A051582, A051583.

List([0..20], n-> Product([0..n-1], j-> 2*j+9) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+9: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

df:=doublefactorial; seq(df(2*n+7)/df(7), n = 0..20); # G. C. Greubel, Nov 12 2019

Table[2^n*Pochhammer[9/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=1,n-1, 2*j+7) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+9) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+7)!!/7!!.
E.g.f.: 1/(1-2*x)^(9/2).
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+9)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 9)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 9*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 9*x/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - ... - (2*n + 7)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 9*x/(1 - 11*x/(1 - 2*x/(1 - 13*x/(1 - 4*x/(1 - 15*x/(1 - 6*x/(1 - ... - (2*n + 9)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 105 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 147, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 77 - 105 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)), where erfi is the imaginary error function. (End)

cp's OEIS Frontend

A144828 Partial products of successive terms of A017113; a(0)=1.

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A167584 The ED4 array read by antidiagonals.

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A257609 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.

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A051578 a(n) = (2*n+4)!!/4!!, related to A000165 (even double factorials).

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A144827 Partial products of successive terms of A017029; a(0)=1.

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A051580 a(n) = (2*n+6)!!/6!!, related to A000165 (even double factorials).

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A257609 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 2*x + 2.