A000246 -id:A000246 - cp's OEIS frontend

A001902 Successive denominators of Wallis's approximation to Pi/2 (reduced).

1, 1, 3, 9, 45, 75, 175, 1225, 11025, 19845, 43659, 160083, 693693, 1288287, 2760615, 41409225, 703956825, 1329696225, 2807136475, 10667118605, 44801898141, 85530896451, 178837328943, 1371086188563, 11425718238025

Offset: 0

N. J. A. Sloane

			From _Wolfdieter Lang_, Dec 07 2017: (Start)
See the table in A001901 for details.
n = 5: numerator(1*2*2*4*4*6/(1*1*3*3*5*5)) = denominator(384/225) = denominator(128/75) = 75. (End)

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

J. Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
J. 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.

Numerators are A001901. For the unreduced form see A001900(n)/A000246(n+1), n >= 0.

a[n_?EvenQ] := n!!^2/((n - 1)!!^2*(n + 1)); a[n_?OddQ] := (n - 1)!!^2*(n + 1)/n!!^2; Table[a[n] // Denominator, {n, 0, 23}] (* Jean-François Alcover, Jun 19 2013 *)

(2*2*4*4*6*6*8*8*...*2n*2n*...)/(1*3*3*5*5*7*7*9*...*(2n-1)*(2n+1)*...) for n >= 1.
From Wolfdieter Lang, Dec 07 2017: (Start)
1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...; partial products (reduced). Here the denominators with offset 0.
a(n) = denominator(W(n)), for n >= 0, with W(n) = Product_{k=0..n} N(k)/D(k) (reduced), with N(k) = 2*floor((k+1)/2) for k >= 1 and N(0) = 1, and D(k) = 2*floor(k/2) + 1, for k >= 0. (End)
a(n) is the denominator of the continued fraction [1;1,1/2,1/3,...,1/n]. - Thomas Ordowski, Oct 19 2024

A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).

Original entry on oeis.org

1, 2, 4, 16, 64, 128, 256, 2048, 16384, 32768, 65536, 262144, 1048576, 2097152, 4194304, 67108864, 1073741824, 2147483648, 4294967296, 17179869184, 68719476736, 137438953472, 274877906944, 2199023255552

Offset: 0

N. J. A. Sloane

If p is prime, then a(p-2) == - A001902(p-2) (mod p). Cf. A064169 (third comment) and my formula here. Such pseudoprimes are 1467, 7831, ... Primes p such that a(p-2) == - A001902(p-2) (mod p^2) are 5, 45827, ... Cf. A355959, see also A330719 (third comment). - Thomas Ordowski, Oct 19 2024

			From _Wolfdieter Lang_, Dec 07 2017: (Start)
The Wallis numerators (N) and denominators (D) with partial products A(n) = A001900(n) and B(n) = A000246(n+1) in unreduced form, and a(n) and b(n) = A001902(n) in reduced form.
n, k:     0  1  2  3  4   5    6     7      8        9       10 ...
N(k):     1  2  2  4  4   6    6     8      8       10       10 ...
D(k):     1  1  3  3  5   5    7     7      9        9        9 ...
A(n):     1  2  4 16 64 384 2304 18432 147456  1474560 14745600 ...
B(n):     1  1  3  9 45 225 1575 11025  99225   893025  9823275 ...
a(n):     1  2  4 16 64 128  256  2048  16384    32768    65536 ...
b(n):     1  1  3  9 45  75  175  1225  11025    19845    43659 ...
n = 5: numerator(1*2*2*4*4*6/(1*1*3*3*5*5)) = numerator(384/225) = numerator(128/75) = 128. (End)

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

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.
Eric Weisstein's World of Mathematics, Pi
Eric Weisstein's World of Mathematics, Pi Continued Fraction
Index to divisibility sequences

Denominators are A001902. Subsequence of A000079.

a[n_?EvenQ] := n!!^2/((n - 1)!!^2*(n + 1)); a[n_?OddQ] := ((n - 1)!!^2*(n + 1))/n!!^2; Table[a[n] // Numerator, {n, 0, 23}] (* Jean-François Alcover, Jun 19 2013 *)

(2*2*4*4*6*6*8*8*...*2n*2n*...)/(1*3*3*5*5*7*7*9*...*(2n-1)*(2n+1)*...) for n >= 1.
From Wolfdieter Lang, Dec 07 2017: (Start)
1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...; partial products (reduced). Here the numerators with offset 0.
a(n) = numerator(W(n)), for n >= 0, with W(n) = Product_{k=0..n} N(k)/D(k) (reduced), with N(k) = 2*floor((k+1)/2) for k >= 1 and N(0) = 1, and D(k) = 2*floor(k/2) + 1, for k >= 0. (End)
a(n) is the numerator of the continued fraction [1;1,1/2,1/3,...,1/n]. - Thomas Ordowski, Oct 19 2024

A130915 Number of permutations in the symmetric group S_n in which cycle lengths are odd and greater than 1.

Original entry on oeis.org

1, 0, 0, 2, 0, 24, 40, 720, 2688, 42560, 245376, 4072320, 31672960, 569935872, 5576263680, 109492807424, 1290701905920, 27616577064960, 380852962029568, 8845627365089280, 139696582370328576, 3506062524305162240, 62387728088875499520, 1684340707284076756992

Offset: 0

Vladeta Jovovic, Aug 23 2007

			a(3)=2 because we have (123) and (132).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 1..200 from Vincenzo Librandi)

Cf. A000246, A000166.

g:=exp(-x)*sqrt((1+x)/(1-x)): gser:=series(g,x=0,30): seq(factorial(n)*coeff(gser,x,n),n=0..20); # Emeric Deutsch, Aug 25 2007
# second Maple program:
a:= proc(n) option remember;
      `if`(n<3, 1/2, a(n-2)+a(n-3))*(n-1)*(n-2)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 18 2024

nn=20;Drop[Range[0,nn]!CoefficientList[Series[((1+x)/(1-x))^(1/2)Exp[-x],{x,0,nn}],x],1]  (* Geoffrey Critzer, Dec 15 2012 *)
a[n_] := (-1)^n*Sum[If[n==k, 1, (-1)^(n + k)*(n - k)!*Sum[Sum[2^(j - i)*StirlingS1[j, i]*Binomial[n - k - 1, j - 1]/j!, {j, i, n - k}], {i, 1, n - k}]*Binomial[n, k]], {k, 0, n}]; Flatten[Table[a[n], {n, 1, 20}]] (* Detlef Meya, Jan 18 2024 *)

my(x='x+O('x^33)); Vec(serlaplace(exp(-x)*sqrt((1+x)/(1-x)))) \\ Joerg Arndt, Jan 18 2024

E.g.f.: exp(-x)*sqrt((1+x)/(1-x)).
a(n) ~ 2*n^n/exp(n+1). - Vaclav Kotesovec, Oct 08 2013
a(n) = (-1)^n*Sum_{k = 0..n} (1 if n = k, otherwise (-1)^(n + k)*(n - k)!*Sum_{i = 1..n - k} Sum_{j = i..n - k} 2^(j - i)*Stirling1(j, i)*binomial(n - k - 1, j - 1)/j!*binomial(n, k)). - Detlef Meya, Jan 18 2024
a(n) = (n-1)*(n-2)*(a(n-2)+a(n-3)) for n>=3. - Alois P. Heinz, Jan 18 2024

More terms from Emeric Deutsch, Aug 25 2007

a(0)=1 prepended by Alois P. Heinz, Jan 18 2024

A288950 Number of relaxed compacted binary trees of right height at most one with empty initial and final sequence on level 0.

Original entry on oeis.org

1, 0, 1, 2, 15, 140, 1575, 20790, 315315, 5405400, 103378275, 2182430250, 50414138775, 1264936572900, 34258698849375, 996137551158750, 30951416768146875, 1023460181133390000, 35885072600989486875, 1329858572860198631250, 51938365373373313209375

Offset: 0

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. The number of unbounded relaxed compacted binary trees of size n is A082161(n). The number of relaxed compacted binary trees of right height at most one of size n is A001147(n). See the Genitrini et al. and Wallner link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where node 3 is at depth 1 on the right of node 2 and where the node n+1 has a left sibling. See the Wallner link. - Michael Wallner, Apr 20 2017

			Denote by L the leaf and by o nodes. Every node has exactly two out-going edges or pointers. Internal edges are denoted by - or |. Pointers are omitted and may point to any node further right. The root is at level 0 at the very left.
The general structure is
  L-o-o-o-o-o-o-o-o-o
    |       |     | |
    o   o-o-o   o-o o.
For n=0 the a(0)=1 solution is L.
For n=1 we have a(1)=0 because we need nodes on level 0 and level 1.
For n=2 the a(2)=1 solution is
     L-o
       |
       o
and the pointers of the node on level 1 both point to the leaf.
For n=3 the a(3)=2 solutions have the structure
     L-o
       |
     o-o
where the pointers of the last node have to point to the leaf, but the pointer of the next node has 2 choices: the leaf of the previous node.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A177145, A213527, A288950, A288952, A288953, A288954 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A001879.

terms = 21; (z + (1 - z)/3*(2 - z + (1 - 2z)^(-1/2)) + O[z]^terms // CoefficientList[#, z] &) Range[0, terms-1]! (* Jean-François Alcover, Dec 04 2018 *)

E.g.f.: z + (1-z)/3 * (2-z + (1-2*z)^(-1/2)).
From Seiichi Manyama, Apr 26 2025: (Start)
a(n) = (n-1)*(2*n-3)/(n-2) * a(n-1) for n > 3.
a(n) = A001879(n-2)/3 for n > 2. (End)

A290383 Number of set partitions of [n] such that the smallest element of each block is odd.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 17, 56, 151, 584, 1893, 8360, 31499, 155720, 666169, 3633704, 17351967, 103284296, 543441005, 3499082408, 20079329875, 138860069192, 861908850561, 6364334129192, 42439075349543, 332934707138888, 2371469004695797, 19681714722718376

Offset: 0

Alois P. Heinz, Jul 28 2017

nonn

a(n) + n is odd for all n > 1.

			a(3) = 2: 123, 12|3.
a(4) = 3: 1234, 124|3, 12|34.
a(5) = 8: 12345, 1234|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 12|34|5.
a(6) = 17: 123456, 12346|5, 1234|56, 12456|3, 1245|36, 1246|35, 124|356, 1246|3|5, 124|36|5, 124|3|56, 1256|34, 125|346, 126|345, 12|3456, 126|34|5, 12|346|5, 12|34|56.

Alois P. Heinz, Table of n, a(n) for n = 0..607
Wikipedia, Partition of a set

Cf. A000110, A000246, A290384.

Bisections give: A307375 (even part), A363589 (odd part).

b:= proc(n, m, t) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), 1-t), j=1..m+1-t))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, m, t) option remember; `if`(n=0, 1,
     `if`(t=0, b(n-1, m+1, 1-t), 0)+m*b(n-1, m, 1-t))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 06 2022

b[n_, m_, t_]:=b[n, m, t]=If[n==0, 1, Sum[b[n - 1, Max[m, j], 1 - t], {j, m + 1 - t}]]; Table[b[n, 0, 0], {n, 0, 50}] (* Indranil Ghosh, Jul 29 2017, after Maple code *)

A288952 Number of relaxed compacted binary trees of right height at most one with empty sequences between branch nodes on level 0.

Original entry on oeis.org

1, 0, 1, 2, 15, 92, 835, 8322, 99169, 1325960, 19966329, 332259290, 6070777999, 120694673748, 2594992240555, 59986047422378, 1483663965460545, 39095051587497488, 1093394763005554801, 32347902448449172530, 1009325655965539561231, 33125674098690460236620

Offset: 0

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels a maximal young leaf has to be followed by a non-maximal young leaf. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

			See A288950 and A288953.

Muniru A Asiru, Table of n, a(n) for n = 0..100
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A177145, A213527, A288950, A288953, A288954 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).

a := [1,0];; for n in [3..10^2] do a[n] := (n-2)*a[n-1] + (n-2)^2*a[n-2]; od; a; # Muniru A Asiru, Jan 26 2018

a:=proc(n) option remember: if n=0 then 1 elif n=1 then 0 elif n>=2 then (n-1)*procname(n-1)-(n-1)^2*procname(n-2) fi; end:
seq(a(n),n=0..100); # Muniru A Asiru, Jan 26 2018

Fold[Append[#1, (#2 - 1) Last[#1] + #1[[#2 - 1]] (#2 - 1)^2] &, {1, 0}, Range[2, 21]] (* Michael De Vlieger, Jan 28 2018 *)

E.g.f.: exp( -Sum_{n>=1} Fibonacci(n-1)*x^n/n ), where Fibonacci(n) = A000045(n).
E.g.f.: exp( -1/sqrt(5)*arctanh(sqrt(5)*z/(2-z)) )/sqrt(1-z-z^2).
a(0) = 1, a(1) = 0, a(n) = (n-1)*a(n-1) + (n-1)^2*a(n-2). - Daniel Suteu, Jan 25 2018

A059838 Number of permutations in the symmetric group S_n that have even order.

Original entry on oeis.org

0, 0, 1, 3, 15, 75, 495, 3465, 29295, 263655, 2735775, 30093525, 370945575, 4822292475, 68916822975, 1033752344625, 16813959537375, 285837312135375, 5214921734397375, 99083512953550125, 2004231846526284375, 42088868777051971875, 934957186489800849375

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Feb 25 2001

From Bob Beals: Let P[n] = probability that a random permutation in S_n has odd order. Then P[n] = sum_k P[random perm in S_n has odd order | n is in a cycle of length k] * P[n is in a cycle of length k]. Now P[n is in a cycle of length k] = 1/n; P[random perm in S_n has odd order | k is even] = 0; P[random perm in S_n has odd order | k is odd] = P[ random perm in S_{n-k} has odd order]. So P[n] = (1/n) * sum_{k odd} P[n-k] = (1/n) P[n-1] + (1/n) sum_{k odd and >=3} P[n-k] = (1/n)*P[n-1] + ((n-2)/n)*P[n-2] and P[1] = 1, P[2] = 1/2. The solution is: P[n] = (1 - 1/2) (1 - 1/4) ... (1-1/(2*[n/2])).

			A permutation in S_4 has even order iff it is a transposition, a product of two disjoint transpositions or a 4 cycle so a(4) = C(4,2)+ C(4,2)/2 + 3! = 15.

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

Cf. A001189, A000246.

List([1..9],n->Length(Filtered(SymmetricGroup(n),x->(Order(x) mod 2)=0)));

s := series((1-sqrt(1-x^2))/(1-x), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

a[n_] := a[n] = n! - ((n-1)! - a[n-1]) * (n+Mod[n, 2]-1); a[0] = 0; Table[a[n], {n, 0, 20}](* Jean-François Alcover, Nov 21 2011, after Pari *)
With[{nn=20},CoefficientList[Series[(1-Sqrt[1-x^2])/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 05 2015 *)

a(n)=if(n<1,0,n!-((n-1)!-a(n-1))*(n+n%2-1))

E.g.f.: (1-sqrt(1-x^2))/(1-x).
a(2n) = (2n-1)! + (2n-1)a(2n-1), a(2n+1) = (2n+1)a(2n).
a(n) = n! - A000246(n). - Victor S. Miller

Additional comments and more terms from Victor S. Miller, Feb 25 2001

Further terms and e.g.f. from Vladeta Jovovic, Feb 28 2001

A060523 Triangle T(n,k) = number of degree-n permutations with k even cycles, k=0..n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 3, 3, 0, 0, 9, 12, 3, 0, 0, 45, 60, 15, 0, 0, 0, 225, 345, 135, 15, 0, 0, 0, 1575, 2415, 945, 105, 0, 0, 0, 0, 11025, 18480, 9030, 1680, 105, 0, 0, 0, 0, 99225, 166320, 81270, 15120, 945, 0, 0, 0, 0, 0, 893025, 1596105, 897750, 217350, 23625, 945, 0, 0, 0, 0, 0

Offset: 0

Vladeta Jovovic, Apr 01 2001

			Triangle T(n,k) begins:
       1;
       1,       0;
       1,       1,      0;
       3,       3,      0,      0;
       9,      12,      3,      0,     0;
      45,      60,     15,      0,     0,   0;
     225,     345,    135,     15,     0,   0, 0;
    1575,    2415,    945,    105,     0,   0, 0, 0;
   11025,   18480,   9030,   1680,   105,   0, 0, 0, 0;
   99225,  166320,  81270,  15120,   945,   0, 0, 0, 0, 0;
  893025, 1596105, 897750, 217350, 23625, 945, 0, 0, 0, 0, 0;
  ...

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 189, Exercise 3.3.13.

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

Columns k=0-1 give: A000246, A096471.
Row sums give A000142.
Cf. A060524, A092691.

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)=0, 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

nn = 6; Range[0, nn]! CoefficientList[
   Series[(1 - x^2)^(-y/2) ((1 + x)/(1 - x))^(1/2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 27 2012 *)

E.g.f.: (1+x)^((1-y)/2)/(1-x)^((1+y)/2).

Sum_{k=0..n} k * T(n,k) = A092691(n). - Alois P. Heinz, Aug 17 2023

A092798 Numerator of partial products in an approximation of Pi/2.

Original entry on oeis.org

2, 16, 8192, 274877906944, 5070602400912917605986812821504, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 1

Ralf Stephan, Mar 05 2004

			The first approximations are 2^(1/2), (16/3)^(1/4), (8192/243)^(1/8), (274877906944/215233605)^(1/16).

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270; arXiv:math/0506319 [math.NT], 2005-2006.
J. Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
J. 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.

Denominators are in A092799.

Cf. A000246, A001900, A001901, A001902, A122214, A122216.

for(m=1, 7, p=1; for(n=1, m, p=p*p*(prod(k=1, ceil(n/2), (2*k)^binomial(n, 2*k-1))/(prod(k=1, floor(n/2)+1, (2*k-1)^binomial(n, 2*k-2))))); print1(numerator(p), ", "))

a(n) = Product_{k=1..n+1} A122214(k)^2^(n-k+1). - Jonathan Sondow, Sep 13 2006

a(n) = Numerator(Product_{k=1..n+1} (A122216(k)/A122217(k))^2^(n-k+1)). - Jonathan Sondow, Sep 13 2006

A102736 Number of permutations of n elements without cycles whose length is a multiple of 3.

Original entry on oeis.org

1, 1, 2, 4, 16, 80, 400, 2800, 22400, 179200, 1792000, 19712000, 216832000, 2818816000, 39463424000, 552487936000, 8839806976000, 150276718592000, 2554704216064000, 48539380105216000, 970787602104320000, 19415752042086400000, 427146544925900800000, 9824370533295718400000, 225960522265801523200000, 5649013056645038080000000, 146874339472770990080000000, 3818732826292045742080000000

Offset: 0

Vladeta Jovovic, Feb 08 2005

Differs from A247007 first at n=27. - Alois P. Heinz, Sep 09 2014

			G.f. = 1 + x + 2*x^2 + 4*x^3 + 16*x^4 + 80*x^5 + 400*x^6 + 2800*x^7 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000090, A000246.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      irem(j, 3)=0, 0, a(n-j)*(j-1)!*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 31 2017

nn=21;a=Sum[x^n/n,{n,3,nn,3}];Range[0,nn]!CoefficientList[Series[Exp[Log[1/(1-x)]-a],{x,0,nn}],x]  (* Geoffrey Critzer, Nov 11 2012 *)
a[ n_] := If[ n < 0, 0, n! With[{m = Quotient[n, 3]}, (-1)^m Binomial[-2/3, m]]]; (* Michael Somos, Aug 05 2016 *)

{a(n) = my(m); if( n<0, 0, m = n\3; n! * (-1)^m * binomial(-2/3, m))}; /* Michael Somos, Aug 05 2016 */

E.g.f.: (1-x^3)^(1/3)/(1-x).

a(n) ~ n! * 3^(1/3) / (GAMMA(2/3) * n^(1/3)). - Vaclav Kotesovec, Mar 15 2014

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A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).

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A130915 Number of permutations in the symmetric group S_n in which cycle lengths are odd and greater than 1.

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A288952 Number of relaxed compacted binary trees of right height at most one with empty sequences between branch nodes on level 0.

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