A000165 -id:A000165 - cp's OEIS frontend

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

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

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)

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)

A137988 Decimal expansion of the number whose Pierce expansion has the sequence of double factorial numbers (A000165) as coefficients.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Feb 26 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric W. Weisstein, Pierce Expansion.
Eric W. Weisstein, Engel Expansion.

Cf. A000142, A137986, A137987, A137989.

P:=proc(n) local a,i,j,k,w; a:=0; w:=1; for i from 0 by 1 to n do k:=i; j:=i-2; while j>0 do k:=k*j; j:=j-2; od; if (i=0 or i=1) then k:=1; fi; if i=2 then k:=2; fi; w:=w*k; a:=a+(-1)^i/w; print(evalf(a,100)); od; end: P(100);

RealDigits[N[(Sum[(-1)^n*Product[1/((k + 1)!!), {k, 1, n}], {n, 1, 250}]), 100]][[1]] (* G. C. Greubel, Jan 01 2017 *)

A137989 Decimal expansion of the inverse of the number whose Engel expansion has the sequence of double factorial numbers (A000165) as coefficients.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Feb 26 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric W. Weisstein, Pierce Expansion.
Eric W. Weisstein, Engel Expansion.

Cf. A000142, A137986, A137987, A137988.

P:=proc(n) local a,i,j,k,w; a:=0; w:=1; for i from 0 by 1 to n do k:=i; j:=i-2; while j>0 do k:=k*j; j:=j-2; od; if (i=0 or i=1) then k:=1; fi; if i=2 then k:=2; fi; w:=w*k; a:=a+1/w; print(evalf(1/a,100)); od; end: P(100);

RealDigits[N[(1/Sum[Product[1/((k - 1)!!), {k, 1, n}], {n, 1, 250}]), 100]][[1]] (* G. C. Greubel, Jan 01 2016 *)

A052289 Numbers n such that n!*2^n + n + 1 = A000165(n) + n + 1 is prime.

Original entry on oeis.org

0, 2, 4, 12, 18, 36

Offset: 1

Labos Elemer, Feb 08 2000

For comparison, A000165(n)+1 is prime for n = 0, 1, 259, 16708, 18655,... (A256594).

a(6) > 25000, if it exists. - Giovanni Resta, Jun 01 2018

			n=4, 2^4*4! + 4 + 1 = 389 is prime.

Cf. A000165, A052290, A256594.

Select[Range[100], PrimeQ[#! 2^# + # + 1] &] (* Giovanni Resta, May 31 2018 *)

a(1)=0 inserted by Sean A. Irvine, Nov 03 2021

A131980 A coefficient tree from the list partition transform relating A000129, A000142, A000165, A110327, and A110330.

Original entry on oeis.org

1, 2, 6, 2, 24, 24, 120, 240, 24, 720, 2400, 720, 5040, 25200, 15120, 720, 40320, 282240, 282240, 40320, 362880, 3386880, 5080320, 1451520, 40320, 3628800, 43545600, 91445760, 43545600, 3628800, 39916800, 598752000, 1676505600, 1197504000, 199584000, 3628800

Offset: 0

Tom Copeland, Oct 30 2007, Nov 29 2007, Nov 30 2007

Construct the infinite array of polynomials
a(0,t) = 1
a(1,t) = 2
a(2,t) = 6 + 2 t
a(3,t) = 24 + 24 t
a(4,t) = 120 + 240 t + 24 t^2
a(5,t) = 720 + 2400 t + 720 t^2
a(6,t) = 5040 + 25200 t + 15120 t^2 + 720 t^3
This array is the reciprocal array of the following array b(n,t) under the list partition transform and its associated operations described in A133314.
b(0,t) = 1, b(1,t) = -2, b(2,t) = -2*(t-1), b(n,t) = 0 for n>2.
Then A000165(n) = a(n,1).
Lower triangular matrix A110327 = binomial(n,k)*a(n-k,2).
n! * A000129(n+1) = a(n,2) = A110327(n,0).
A110330 = matrix inverse of binomial(n,k)*a(n-k,2) = binomial(n,k)*b(n-k,2).
A000142(n+1) = a(n,0).
From Peter Bala, Sep 09 2013: (Start)
Let {P(n,x)}n>=0 be a polynomial sequence. Koutras has defined generalized Eulerian numbers associated with the sequence P(n,x) as the coefficients A(n,k) in the expansion of P(n,x) in a series of factorials of degree n, namely P(n,x) = Sum_{k=0..n} A(n,k)* binomial(x+n-k,n). The choice P(n,x) = x^n produces the classical Eulerian numbers of A008292. Let now P(n,x) = x*(x + 1)*...*(x + n - 1) denote the n-th rising factorial polynomial. Then the present table is the generalized Eulerian numbers associated with the polynomial sequence P(n,2*x). See A228955 for the generalized Eulerian numbers associated with the polynomial sequence P(n,2*x + 1). (End)

			Triangle begins as:
        1;
        2;
        6,        2;
       24,       24;
      120,      240,       24;
      720,     2400,      720;
     5040,    25200,    15120,      720;
    40320,   282240,   282240,    40320;
   362880,  3386880,  5080320,  1451520,   40320;
  3628800, 43545600, 91445760, 43545600, 3628800;

M. V. Koutras, Eulerian numbers associated with sequences of polynomials, The Fibonacci Quarterly, 32 (1994), 44-57.

Cf. A228955.

Flat(List([0..10], n-> List([0..Int(n/2)], k-> Factorial(n)*Binomial(n+1, 2*k+1) ))); # G. C. Greubel, Dec 30 2019

[Factorial(n)*Binomial(n+1, 2*k+1): k in [0..Floor(n/2)], n in [0..10]]; // G. C. Greubel, Dec 30 2019

for n from 0 to 10 do
seq( n!*binomial(n+1,2*k+1), k = 0..floor(n/2) )
end do; # Peter Bala, Sep 09 2013

Table[n!*Binomial[n+1, 2*k+1], {n,0,10}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Dec 30 2019 *)

T(n,k) = n!*binomial(n+1, 2*k+1);
for(n=0,10, for(k=0, n\2, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 30 2019

[[factorial(n)*binomial(n+1, 2*k+1) for k in (0..floor(n/2))] for n in (0..10)] # G. C. Greubel, Dec 30 2019

E.g.f. for the polynomials b(.,t), introduced above, is 1 - 2x - (t-1) * x^2; therefore e.g.f. for the polynomials a(.,t), which are the row polynomials of this array, is 1 / ( 1 - 2x - (t-1) * x^2 ) = (t-1) / ( t - ( 1 + x*(t-1) )^2 ).
Also, a(n,t) = (1 - t*u^2)^(n+1) (D_u)^n [ 1 / (1 - t*u^2) ] with eval. at u = 1/t. Compare A076743.
a(n,t) = n!*Sum_{k>=0} binomial(n+1,2k+1) * t^k = n!*Sum_{k>=0} A034867(n,k) * t^k.
Additional relations are given by formulas in A133314.
From Peter Bala, Sep 09 2013: (Start)
Recurrence equation: T(n+1,k) = (n+2 +2*k)T(n,k) + (n +2 -2*k)T(n,k-1).
Let P(n,x) = x*(x + 1)*...*(x + n - 1) denote the n-th rising factorial.
T(n,k) = Sum_{j=0..k+1} (-1)^(k+1-j)*binomial(n+1,k+1-j)*P(n,2*j) for n >= 1.
The row polynomial a(n,t) satisfies t*a(n,t)/(1 - t)^(n+1) = Sum_{j>=1} P(n,2*j)*t^j. For example, for n = 3 we have t*(24 + 24*t)/(1 - t)^4 = 2*3*4*t + (4*5*6)*t^2 + (6*7*8)*t^3 + ..., while for n = 4 we have t*(120 + 240*t + 24*t^2)/(1 - t)^5 = (2*3*4*5)*t + (4*5*6*7)*t^2 + (6*7*8*9)*t^3 + .... (End)

Removed erroneous and duplicate statements. - Tom Copeland, Dec 03 2013

A208057 Triangle by rows, generated from the odd integers and related to A000165.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 24, 18, 5, 1, 192, 144, 40, 7, 1, 1920, 1440, 400, 70, 9, 1, 23040, 17280, 4800, 840, 108, 11, 1, 322560, 241920, 67200, 11760, 1512, 154, 13, 1, 5160960, 3870720, 1075200, 188160, 24192, 2464, 208, 15, 1

Offset: 0

Gary W. Adamson, Feb 22 2012

Row sums = A000165, the double factorial numbers: (1, 2, 8, 48, 384,...).

Left border = A002866 and the eigensequence of the odd integers prefaced with a 1.

			First few rows of the triangle:
  1;
  1, 1;
  4, 3, 1;
  24, 18, 5, 1;
  192, 144, 40, 7, 1;
  1920, 1440, 400, 70, 9, 1;
  23040, 17280, 4800, 840, 108, 11, 1;
  322560, 241920, 67200, 11760, 1512, 154, 13, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000165, A002866, A158405.

T:= proc(n) option remember; local M;
      M:= (Matrix(n+1, (i, j)-> `if`(i=j, 1, `if`(i>j, -2*j+1, 0)))^(-1));
      seq(M[n+1, k], k=1..n+1)
    end:
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 27 2012

T[n_] := T[n] = Module[{M}, M = Table[If[i == j, 1, If[i>j, -2*j+1, 0]], {i, 1, n+1 }, {j, 1, n+1}] // Inverse; M[[n+1]]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

Eigentriangle of triangle A158405 (odd integers in every row: (1, 3, 5,...)); the inverse of:
   1;
  -1,  1;
  -1, -3,  1;
  -1, -3, -5,  1;
  -1, -3, -5, -7,  1;
  ...

Typo in term 17 corrected by Alois P. Heinz, Dec 06 2012

A136807 Hankel transform of double factorial numbers n!*2^n=A000165(n).

Original entry on oeis.org

1, 4, 256, 589824, 86973087744, 1282470362637926400, 2723154477021188283432960000, 1133321924829207204666583887642624000000, 120746421332702772771144114237731253721340313600000000

Offset: 0

Paul Barry, Jan 23 2008

By the properties of the Hankel transform, a(n)=2^(n(n+1))*A055209(n).

Also Hankel transform of A000354, A010844, A082032. - Philippe Deléham, Jan 23 2008

G. C. Greubel, Table of n, a(n) for n = 0..28
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000354, A010844, A082032, A055209.

[1] cat [(&*[(2*k)^(2*(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[(2k)^(2(n-k+1)),{k,n}],{n,0,10}] (* Harvey P. Dale, Apr 11 2013 *)

for(n=0,10, print1(prod(k=1,n,(2*k)^(2*(n-k+1))), ", ")) \\ G. C. Greubel, Oct 14 2018

a(n) = Product_{k=1..n} (2k)^(2(n-k+1)).

a(n) ~ 2^((n+1)^2) * Pi^(n+1) * n^(n^2 + 2*n + 5/6) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Feb 24 2019

A052290 Prime values of n!*2^n+n+1 = A000165(n)+n+1.

Original entry on oeis.org

2, 11, 389, 1961990553613, 1678343852714360832019, 25563186766285862273530264901662157745369907200000037

Offset: 1

Labos Elemer, Feb 08 2000

nonn

For a(6), n = 36. There are no additional terms up to n=1000. - Harvey P. Dale, Aug 06 2012

Primes for n = 0, 2, 4, 12, 18, 36, and no others for n < 5001. - Robert G. Wilson v, Aug 07 2012

			If n=4, 2^4*4!+4+1 = 389 is prime, so 389 is a term.

Cf. A000165, A052289.

Select[Table[n! 2^n+n+1,{n,0,1000}],PrimeQ] (* Harvey P. Dale, Aug 06 2012 *)

cp's OEIS Frontend

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

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

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

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A137988 Decimal expansion of the number whose Pierce expansion has the sequence of double factorial numbers (A000165) as coefficients.

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A137989 Decimal expansion of the inverse of the number whose Engel expansion has the sequence of double factorial numbers (A000165) as coefficients.

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Maple

Mathematica

A052289 Numbers n such that n!*2^n + n + 1 = A000165(n) + n + 1 is prime.

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A131980 A coefficient tree from the list partition transform relating A000129, A000142, A000165, A110327, and A110330.

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