A052319 -id:A052319 - cp's OEIS frontend

A006882 Double factorials n!!: a(n) = n*a(n-2) for n > 1, a(0) = a(1) = 1.

1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, 185794560, 654729075, 3715891200, 13749310575, 81749606400, 316234143225, 1961990553600, 7905853580625, 51011754393600, 213458046676875, 1428329123020800

Offset: 0

Robert Munafo

Product of pairs of successive terms gives factorials in increasing order. - Amarnath Murthy, Oct 17 2002
a(n) = number of down-up permutations on [n+1] for which the entries in the even positions are increasing. For example, a(3)=3 counts 2143, 3142, 4132. Also, a(n) = number of down-up permutations on [n+2] for which the entries in the odd positions are decreasing. For example, a(3)=3 counts 51423, 52413, 53412. - David Callan, Nov 29 2007
The double factorial of a positive integer n is the product of the positive integers <= n that have the same parity as n. - Peter Luschny, Jun 23 2011
For n even, a(n) is the number of ways to place n points on an n X n grid with pairwise distinct abscissa, pairwise distinct ordinate, and 180-degree rotational symmetry. For n odd, the number of ways is a(n-1) because the center point can be considered "fixed". For 90-degree rotational symmetry cf. A001813, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017
Could be extended to include a(-1) = 1. But a(-2) is not defined, otherwise we would have 1 = a(0) = 0*a(-2). - Jianing Song, Oct 23 2019

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 48*x^6 + 105*x^7 + 384*x^8 + ...

Putnam Contest, 4 Dec. 2004, Problem A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 0..806 (terms 0..100 from T. D. Noe)
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
CombOS - Combinatorial Object Server, Generate colored permutations
Joseph E. Cooper III, A recurrence for an expression involving double factorials, arXiv:1510.00399 [math.CO], 2015.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Gary T. Leavens and Mike Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
Peter Luschny, Multifactorials
B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425-426.
Rudolph Ondrejka, Tables of double factorials, Math. Comp., Vol. 24, No. 109 (1970), p. 231.
Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, Multifactorial.
Index entries for sequences related to factorial numbers
Index entries for "core" sequences

Bisections are A000165 and A001147. These two entries have more information.
Cf. A052319, A143280.
A diagonal of A202212.
Cf. A000085, A001813, A135401, A297708. - Manfred Scheucher, Jan 07 2018

a006882 n = a006882_list !! n
a006882_list = 1 : 1 : zipWith (*) [2..] a006882_list
-- Reinhard Zumkeller, Oct 23 2014

DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(n): n in [0..28] ]; // Klaus Brockhaus, Jan 23 2011

A006882 := proc(n) option remember; if n <= 1 then 1 else n*A006882(n-2); fi; end;
A006882 := proc(n) doublefactorial(n) ; end proc; seq(A006882(n),n=0..10) ; # R. J. Mathar, Oct 20 2009
A006882 := n -> mul(k, k = select(k -> k mod 2 = n mod 2, [$1 .. n])):  seq(A006882(n), n = 0 .. 10); # Peter Luschny, Jun 23 2011
A006882 := proc(n) if n=0 then 1 else mul(n-2*k, k=0..floor(n/2)-1); fi; end; # N. J. A. Sloane, May 27 2016

Array[ #!!&, 40, 0 ]
multiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*multiFactorial[n - k, k]]]; Array[ multiFactorial[#, 2] &, 27, 0] (* Robert G. Wilson v, Apr 23 2011 *)

{a(n) = prod(i=0, (n-1)\2, n - 2*i )} \\ Improved by M. F. Hasler, Nov 30 2013

{a(n) = if( n<2, n>=0, n * a(n-2))}; /* Michael Somos, Apr 06 2003 */

{a(n) = if( n<0, 0, my(E); E = exp(x^2 / 2 + x * O(x^n)); n! * polcoeff( 1 + E * x * (1 + intformal(1 / E)), n))}; /* Michael Somos, Apr 06 2003 */

from sympy import factorial2
def A006882(n): return factorial2(n) # Chai Wah Wu, Apr 03 2021

a(n) = Product_{i=0..floor((n-1)/2)} (n - 2*i).
E.g.f.: 1+exp(x^2/2)*x*(1+sqrt(Pi/2)*erf(x/sqrt(2))). - Wouter Meeussen, Mar 08 2001
Satisfies a(n+3)*a(n) - a(n+1)*a(n+2) = (n+1)!. [Putnam Contest]
a(n) = n!/a(n-1). - Vaclav Kotesovec, Sep 17 2012
a(n) * a(n+3) = a(n+1) * (a(n+2) + a(n)). a(n) * a(n+1) = (n+1)!. - Michael Somos, Dec 29 2012
a(n) ~ c * n^((n+1)/2) / exp(n/2), where c = sqrt(Pi) if n is even, and c = sqrt(2) if n is odd. - Vaclav Kotesovec, Nov 08 2014
a(2*n) = 2^n*a(n)*a(n-1). a(2^n) = 2^(2^n - 1) * 1!! * 3!! * 7!! * ... * (2^(n-1) - 1)!!. - Peter Bala, Nov 01 2016
a(n) = 2^h*(2/Pi)^(sin(Pi*h)^2/2)*Gamma(h+1) where h = n/2. This analytical extension supports the view that a(-1) = 1 is a meaningful numerical extension. With this definition (-1/2)!! = Gamma(3/4)/Pi^(1/4). - Peter Luschny, Oct 24 2019
a(n) ~ (n+1/6)*sqrt((2/e)*(n/e)^(n-1)*(Pi/2)^(cos(n*Pi/2)^2)). - Peter Luschny, Oct 25 2019
Sum_{n>=0} 1/a(n) = A143280. - Amiram Eldar, Nov 10 2020
Sum_{n>=0} 1/(a(n)*a(n+1)) = e - 1. - Andrés Ventas, Apr 12 2021

A052318 Number of labeled rooted trimmed trees with n nodes.

Original entry on oeis.org

1, 2, 3, 16, 145, 1536, 19579, 290816, 4942305, 94689280, 2020278931, 47523053568, 1222147737265, 34117226135552, 1027550555918475, 33213871550365696, 1146891651823112641, 42135941698113503232, 1641164216596258397347, 67550839668807638712320

Offset: 1

Christian G. Bower, Dec 15 1999

A rooted trimmed tree is a tree with a forbidden limb of length 2.

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Alois P. Heinz, Table of n, a(n) for n = 1..400
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052319-A052329.

A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)-x^2), x,n), polynom) fi end: a:= n-> coeff(A(n+1), x,n)*n!: seq(a(n), n=1..25); # Alois P. Heinz, Aug 23 2008

a[n_] := Sum[ Boole[ EvenQ[n-m]]*(m^((n+m)/2-2)/((n-m)/2)!)*((-1)^((n-m)/2)/(m-1)!), {m, 1, n}]*n!; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Sep 10 2012, after Vladimir Kruchinin *)
Rest[CoefficientList[Series[-LambertW[-x/E^(x^2)],{x,0,20}],x]*Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

a(n):=sum((if mod(n-m,2)=0 then m^((n+m)/2-2)/((n-m)/2)!*(-1)^((n-m)/2) else 0)/(m-1)!,m,1,n); /* Vladimir Kruchinin, Aug 07 2012 */

E.g.f. satisfies A(x) = x*exp(A(x) - x^2).
E.g.f.: -LambertW(-x/exp(x^2)). - Vaclav Kotesovec, Jan 08 2014
a(n) ~ sqrt(1 + LambertW(-2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n) * (-LambertW(-2*exp(-2)))^(n/2)). - Vaclav Kotesovec, Jan 08 2014

A240885 Decimal expansion of the unique positive solution of Integral_{0..x} exp(-t^2/2) dt = 1.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 14 2014

			1.2755477364172...

Steven Finch, Pattern-Avoiding Permutations [Internet archive]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]

Cf. A052319, A071075, A111004, A200404, A263885.

RealDigits[Sqrt[2]*InverseErf[Sqrt[2/Pi]], 10, 100] // First

a=sqrt(Pi/2); b=a-1; c=1/sqrt(2); solve(x=1,2, a*erfc(c*x)-b) \\ Charles R Greathouse IV, Sep 02 2024

Solution to sqrt(Pi/2)*erf(x/sqrt(2)) = 1.

A052322 Number of labeled rooted trees with a forbidden limb of length 3.

Original entry on oeis.org

1, 2, 9, 40, 385, 4536, 66409, 1127792, 21981537, 483858640, 11873508361, 321497975448, 9522483900241, 306292854886760, 10632656242583145, 396223803663328096, 15776491521834720961, 668460175137505993248, 30030668624358362706697, 1425868954034374729854920

Offset: 1

Christian G. Bower, Dec 15 1999

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Alois P. Heinz, Table of n, a(n) for n = 1..390
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052319-A052329.

Rest[CoefficientList[Series[-LambertW[-x*E^(-x^3)], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

E.g.f. satisfies A(x) = x*exp(A(x) - x^3).
E.g.f: -LambertW(-x*exp(-x^3)). - Vaclav Kotesovec, Jan 08 2014
a(n) ~ sqrt(1+LambertW(-3*exp(-3))) * n^(n-1) * exp(n/3*LambertW(-3*exp(-3))). - Vaclav Kotesovec, Jan 08 2014

A052324 Number of increasing rooted trees with a forbidden limb of length 3.

Original entry on oeis.org

0, 1, 1, 2, 5, 19, 90, 520, 3475, 26550, 228050, 2177020, 22860090, 261870070, 3249793360, 43432062300, 621911561150, 9498946124800, 154152712434600, 2648808048264400, 48043086765929200, 917249983543337400

Offset: 0

Christian G. Bower, Dec 15 1999

In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Vaclav Kotesovec, Table of n, a(n) for n = 0..240
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052319-A052329.

CoefficientList[Assuming[{Element[x, Reals], x > 0}, Series[-Log[1-6^(1/3)*Gamma[1/3]/3 + 1/3*x*ExpIntegralE[2/3, x^3/6]], {x, 0, 20}]], x]*Range[0, 20]! (* Vaclav Kotesovec, Mar 28 2014 *)

{a(n)=local(A=x); for(i=0, n, A=intformal(exp(A-x^3/6+O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 28 2014

E.g.f. satisfies A'(x) = exp(A(x) - x^3/6). - corrected by Vaclav Kotesovec, Mar 28 2014
a(n) ~ d^n * (n-1)!, where d = 0.9546118344740519430556804... - Vaclav Kotesovec, Mar 28 2014
In closed form, d = 1/r, where r = 1.04754620033697244977759528695194261... is the root of the equation 1 = Integral_{x=0..r} exp(-x^3/6) dx. - Vaclav Kotesovec, Aug 21 2014

A002990 Number of n-node trees with a forbidden limb of length 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 9, 19, 38, 86, 188, 439, 1026, 2472, 5997, 14835, 36964, 93246, 236922, 607111, 1565478, 4062797, 10599853, 27797420, 73224806, 193709710, 514406793, 1370937140, 3665714528, 9831891555, 26445886506, 71325268179

Offset: 0

N. J. A. Sloane

nonn

A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps.

A. J. Schwenk, Almost all trees are cospectral, pp. 275-307 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972.
Index entries for sequences related to trees

Cf. A002955, A002988, A002989, A002990, A002991, A002992.
Cf. A052318, A052319, A052320.
Cf. A052321, A052322, A052323, A052324, A052325, A052326.
Cf. A052327, A052328, A052329.

with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=4, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= n-> `if`(n=0, 1, g(n-1)+(`if`(irem(n, 2, 'r')=0,
         g(r-1), 0)-add(g(i-1)*g(n-i-1), i=1..n-1))/2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 06 2014

g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 4, 1, 0]), {d, Divisors[j] }]*g[n-j], {j, 1, n}]/n]; a[n_] := If[n == 0, 1, g[n-1] + (If[Mod[n, 2 ] == 0, g[Quotient[n, 2]-1], 0] - Sum[g[i-1]*g[n-i-1], {i, 1, n-1}])/2]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

G.f.: 1 + B(x) + (B(x^2) - B(x)^2)/2 where B(x) is g.f. of A052327.

a(n) ~ c * d^n / n^(5/2), where d = 2.9224691962496551739365155005926..., c = 0.503471518908815272581177797536... . - Vaclav Kotesovec, Aug 25 2014

More terms, formula and comments from Christian G. Bower, Dec 15 1999

cp's OEIS Frontend

A006882 Double factorials n!!: a(n) = n*a(n-2) for n > 1, a(0) = a(1) = 1.

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A052318 Number of labeled rooted trimmed trees with n nodes.

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A240885 Decimal expansion of the unique positive solution of Integral_{0..x} exp(-t^2/2) dt = 1.

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A052322 Number of labeled rooted trees with a forbidden limb of length 3.

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A052324 Number of increasing rooted trees with a forbidden limb of length 3.

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A002990 Number of n-node trees with a forbidden limb of length 4.

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