A000246 -id:A000246 - cp's OEIS frontend

A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

1, 1, 3, 7, 49, 321, 2131, 19783, 195777, 2101249, 25721731, 340358151, 4902173233, 75688032577, 1253701725459, 22347046050631, 418439924732161, 8318748086461953, 175769214730290307, 3871849719998940679, 89734800330818444721, 2187944831367633226561

Offset: 0

Vladeta Jovovic, Jan 19 2006

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A000726, A003724, A115276, A000246, A102736, A088009, A001590.

nmax := 30: B := x*(1+x)/(1-x^3) : egf := 0 : for i from 0 to nmax do egf := convert(egf+taylor(B^i,x=0,nmax+1)/i!,polynom) : od: for i from 0 to nmax do printf("%d ", i!*coeftayl(egf,x=0,i)) ; od: # R. J. Mathar, Feb 06 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(0=
      irem(j, 3), 0, a(n-j)*j!*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

CoefficientList[Series[E^(x*(1+x)/(1-x^3)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)

E.g.f.: exp(x*(1+x)/(1-x^3)).
a(n) = a(n-1) + 2*(n-1)*a(n-2) + 2*(n-3)*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Sep 25 2013
a(n) ~ 6^(-1/4) * n^(n-1/4) * exp(2/3*sqrt(6*n)-n) * (1 - 43/(48*sqrt(6*n))). - Vaclav Kotesovec, Sep 25 2013

2 more terms from R. J. Mathar, Feb 06 2008

A193437 Expansion of e.g.f. exp( Sum_{n>=0} x^(3n+1)/(3n+1) ).

Original entry on oeis.org

1, 1, 1, 1, 7, 31, 91, 931, 7441, 38017, 507241, 5864761, 43501591, 713059711, 10776989587, 105784464331, 2052437475361, 38263122487681, 469863736958161, 10518597616325617, 232980391759702951, 3446848352553524191, 87385257330831947851

Offset: 0

Paul D. Hanna, Jul 25 2011

nonn

Conjecture: a(n) is divisible by 7^floor(n/7) for n>=0.
Conjecture: a(n) is divisible by p^floor(n/p) for prime p == 1 (mod 3).
a(n) is the number of permutations of n elements with a disjoint cycle decomposition in which every cycle length is == 1 (mod 3). - Simon Tatham, Mar 26 2021

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + 91*x^6/6! + 931*x^7/7! +...
where
log(A(x)) = x + x^4/4 + x^7/7 + x^10/10 + x^13/13 + x^16/16 + x^19/19 +...

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000246, A193438.

nmax = 30; CoefficientList[Series[Exp[x*Hypergeometric2F1[1/3, 1, 4/3, x^3]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

{a(n)=n!*polcoeff( exp(sum(m=0,n,x^(3*m+1)/(3*m+1))+x*O(x^n)) ,n)}

a(n) = a(n-1) + (n-3)*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Apr 15 2020
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * (3*k)! * a(n-3*k-1). - Ilya Gutkovskiy, Jul 14 2021
E.g.f.: A(x) = exp(Integral_{z = 0..x} 1/(1-z^3) dz) = (1-x^3)^(1/6)/(1-x)^(1/2) * exp((1/sqrt(3))*arctan(sqrt(3)*x/(2+x))). - Fabian Pereyra, Oct 14 2023

A258830 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums.

Original entry on oeis.org

1, 1, 2, 5, 20, 87, 522, 3271, 26168, 214955, 2149550, 21881103, 262573236, 3191361201, 44679056814, 631546127049, 10104738032784, 162891774138339, 2932051934490102, 53094870211027831, 1061897404220556620, 21342730463672017301, 469540070200784380622

Offset: 0

Alois P. Heinz, Jun 11 2015

nonn

			p = 1432 is counted by a(4) because the up-down signature of 0,p = 01432 is 1,1,-1,-1 with partial sums 1,2,1,0.
a(0) = 1: the empty permutation.
a(1) = 1: 1.
a(2) = 2: 12, 21.
a(3) = 5: 123, 132, 213, 231, 312.
a(4) = 20: 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3241, 3412, 3421, 4123, 4132, 4231.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Row sums of A258829.
Main diagonal of A262163.
Cf. A000246.

b:= proc(u, o, c) option remember;
      `if`(c<0, 0, `if`(u+o<=c, (u+o)!,
       add(b(u-j, o-1+j, c+1), j=1..u)+
       add(b(u+j-1, o-j, c-1), j=1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);

b[u_, o_, c_] := b[u, o, c] = If[c < 0, 0, If[u + o <= c, (u + o)!,
    Sum[b[u - j, o - 1 + j, c + 1], {j, 1, u}] +
    Sum[b[u + j - 1, o - j, c - 1], {j, 1, o}]]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 30] (* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

a(n) ~ c * n! / sqrt(n), where c = 2.03565662136472375868003536175448... . - Vaclav Kotesovec, Jun 21 2015

A288953 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except after the last branch node on level 0.

Original entry on oeis.org

1, 1, 3, 10, 51, 280, 1995, 15120, 138075, 1330560, 14812875, 172972800, 2271359475, 31135104000, 471038042475, 7410154752000, 126906349444875, 2252687044608000, 43078308695296875, 851515702861824000, 17984171447178811875, 391697223316439040000

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 maximal young leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the beginning. 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

			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.
For n=0 the a(0)=1 solution is L.
For n=1 the a(1)=1 solution is L-o.
For n=2 the a(2)=3 solutions are
L-o-o     L-o
            |
            o
  2    +   1    solutions of this shape with pointers.

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. A288954 (variation with additional initial sequence).
Cf. A177145 (variation without final sequence).
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, A213527, A288950, A288952, 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).

E.g.f.: (2-z)/(3*(1-z)^2) + 1/(3*sqrt(1-z^2)).

A321280 Number T(n,k) of permutations p of [n] with exactly k descents such that the up-down signature of p has nonnegative partial sums; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 8, 1, 22, 22, 1, 52, 172, 1, 114, 856, 604, 1, 240, 3488, 7296, 1, 494, 12746, 54746, 31238, 1, 1004, 43628, 330068, 518324, 1, 2026, 143244, 1756878, 5300418, 2620708, 1, 4072, 457536, 8641800, 43235304, 55717312, 1, 8166, 1434318, 40298572, 309074508, 728888188, 325024572

Offset: 0

Alois P. Heinz, Nov 01 2018

			Triangle T(n,k) begins:
  1;
  1;
  1;
  1,     2;
  1,     8;
  1,    22,      22;
  1,    52,     172;
  1,   114,     856,       604;
  1,   240,    3488,      7296;
  1,   494,   12746,     54746,      31238;
  1,  1004,   43628,    330068,     518324;
  1,  2026,  143244,   1756878,    5300418,    2620708;
  1,  4072,  457536,   8641800,   43235304,   55717312;
  1,  8166, 1434318,  40298572,  309074508,  728888188,  325024572;
  1, 16356, 4438540, 180969752, 2026885824, 7589067592, 8460090160;
  ...

Alois P. Heinz, Rows n = 0..100, flattened
S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv preprint arXiv:1810.00993 [math.CO], 2018.
David G. L. Wang, T. Zhao, The peak and descent statistics over ballot permutations, arXiv:2009.05973 [math.CO], 2020.

Columns k=0-3 give: A000012, A005803 (for n>0), A321268, A321269.
Row sums give A000246.
T(2n+1,n) gives A177042.
T(2n+2,n) gives A303285(n+1).
Cf. A262124, A262125.

b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, 1/x,
       add(expand(x*b(u-j, o-1+j, c-1)), j=1..u)+
       add(b(u+j-1, o-j, c+1), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(`if`(n=0, 1, b(n, 0, 1))):
seq(T(n), n=0..14);

b[u_, o_, c_] := b[u, o, c] = If[c < 0, 0, If[u + o == 0, 1/x, Sum[Expand[ x*b[u - j, o - 1 + j, c - 1]], {j, 1, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ If[n == 0, 1, b[n, 0, 1]]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 08 2018, after Alois P. Heinz *)

A002867 a(n) = binomial(n,floor(n/2))*(n+1)!.

Original entry on oeis.org

1, 2, 12, 72, 720, 7200, 100800, 1411200, 25401600, 457228800, 10059033600, 221298739200, 5753767219200, 149597947699200, 4487938430976000, 134638152929280000, 4577697199595520000, 155641704786247680000, 5914384781877411840000, 224746621711341649920000

Offset: 0

N. J. A. Sloane

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).

T. D. Noe, Table of n, a(n) for n = 0..100
Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
Theodore S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers.

Cf. A000246.

Table[Binomial[n,Floor[n/2]](n+1)!,{n,0,20}] (* Harvey P. Dale, Sep 04 2018 *)

a(n) = 2^n * A000246(n+1).
E.g.f.: 1/(sqrt(1+2*x)*(1-2*x)^(3/2)) = 1/(sqrt(1-4*x^2)*(1-2*x)). - Paul Barry, Jul 22 2005
Conjecture: a(n) - 2*a(n-1) - 4*n*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
Sum_{n>=0} 1/a(n) = (StruveL(-1,1/2) + StruveL(0,1/2))*Pi/2, where StruveL is the modified Struve function. - Amiram Eldar, Aug 15 2025

More terms from James Sellers, Jul 10 2000

A053195 Number of level permutations of degree n.

Original entry on oeis.org

1, 1, 2, 3, 18, 45, 360, 1575, 20790, 99225, 1332450, 9823275, 181496700, 1404728325, 26221595400, 273922023375, 7196040101250, 69850115960625, 1662139682453250, 22561587455281875, 675158520854317500, 9002073394657468125, 259715927440434465000

Offset: 0

Vladeta Jovovic, Mar 02 2000

A permutation is level if the powers of 2 dividing its cycle lengths are all equal.

For odd n, level permutations of degree n are just permutations that have odd order, i.e., A053195(2*n+1) = A000246(2*n+1). - Vladeta Jovovic, Sep 29 2004

Alois P. Heinz, Table of n, a(n) for n = 0..400
L. Babai and P. J. Cameron, Automorphisms and enumeration of switching classes of tournaments, Electron. J. Combin., 7 (2000), no. 1, Research Paper 38, 25 pp.

Cf. A049313, A053197.

with(combinat):
b:= proc(n, i, p) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j*
       b(n-i*j, i-2*p, p), j=0..n/i)))
    end:
a:= n-> (m-> `if`(n=0, 1, add(b(n, (h-> h-1+irem(h, 2)
    )(iquo(n, 2^j))*2^j, 2^j), j=0..m)))(ilog2(n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 11 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, p_] := b[n, i, p] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*(i-1)!^j*b[n - i*j, i-2*p, p], {j, 0, n/i}]]]; a[n_] := Function[{m}, If[n == 0, 1, Sum[ b[n, Function [{h}, h - 1 + Mod[h, 2]][Quotient[n, 2^j]]*2^j, 2^j], {j, 0, m}]]][Log[2, n] // Floor]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 07 2015, after Alois P. Heinz *)

a(0)=1 prepended by Alois P. Heinz, Jun 11 2015

A092799 Denominator of partial products in an approximation to Pi/2.

Original entry on oeis.org

1, 3, 243, 215233605, 2849452841966467687734375, 34139907905802495953388390516678108673704867996275424957275390625

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.

Numerators are in A092798.
Cf. A000246, A001900, A001901, A001902.
Cf. A122215, A122217.

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(denominator(p), ", "))

a(n) = Product_{k=1..n+1} A122215(k)^2^(n-k+1). - Jonathan Sondow, Sep 13 2006

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

A193438 Expansion of e.g.f. exp( Sum_{n>=0} x^(4n+1)/(4n+1) ).

Original entry on oeis.org

1, 1, 1, 1, 1, 25, 145, 505, 1345, 43345, 481825, 3027025, 13679425, 528618025, 8796735025, 81517983625, 529655946625, 23619691278625, 526089195906625, 6512769913326625, 55783484692170625, 2802281186570685625, 78369733286598300625, 1221751619270220585625

Offset: 0

Paul D. Hanna, Jul 25 2011

nonn

Conjecture: a(n) is divisible by 5^floor(n/5) for n>=0.

Conjecture: a(n) is divisible by p^floor(n/p) for prime p == 1 (mod 4).

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + 25*x^5/5! + 145*x^6/6! + 505*x^7/7! +...
where
log(A(x)) = x + x^5/5 + x^9/9 + x^13/13 + x^17/17 + x^21/21 + x^25/25 +...

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000246, A193437.

nmax = 30; CoefficientList[Series[Exp[x*Hypergeometric2F1[1/4, 1, 5/4, x^4]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

{a(n)=n!*polcoeff( exp(sum(m=0,n,x^(4*m+1)/(4*m+1))+x*O(x^n)) ,n)}

seq(n) = Vec(serlaplace(exp(intformal(1/(1-x^4) + O(x*x^n)) ))) \\ Andrew Howroyd, Oct 15 2023

a(n) = a(n-1) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Apr 15 2020
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * (4*k)! * a(n-4*k-1). - Ilya Gutkovskiy, Jul 14 2021
E.g.f.: A(x) = exp(Integral_{z = 0..x} 1/(1-z^4) dz) = exp((arctan(x)+arctanh(x))/2). - Fabian Pereyra, Oct 14 2023

A288954 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except before the first and after the last branch node on level 0.

Original entry on oeis.org

1, 1, 3, 13, 79, 555, 4605, 42315, 436275, 4894155, 60125625, 794437875, 11325612375, 172141044075, 2793834368325, 48009995908875, 874143494098875, 16757439016192875, 338309837281040625, 7157757510792763875, 158706419654857449375, 3673441093896736036875

Offset: 2

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 leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the begininning and at the end. 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.

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. A288953 (variation without initial sequence).
Cf. A177145 (variation without initial and final sequence).
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, A213527, A288950, A288952 (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).

terms = 22; egf = 1/(3(1-z))(1/Sqrt[1-z^2] + (3z^3 - z^2 - 2z + 2)/((1-z)(1-z^2))) + O[z]^terms;
CoefficientList[egf, z] Range[0, terms-1]! (* Jean-François Alcover, Dec 13 2018 *)

E.g.f.: 1/(3*(1-z))*( 1/sqrt(1-z^2) + (3*z^3-z^2-2*z+2)/((1-z)*(1-z^2)) ).

cp's OEIS Frontend

A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

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A193437 Expansion of e.g.f. exp( Sum_{n>=0} x^(3*n+1)/(3*n+1) ).

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A258830 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums.

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A288953 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except after the last branch node on level 0.

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A321280 Number T(n,k) of permutations p of [n] with exactly k descents such that the up-down signature of p has nonnegative partial sums; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.

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A002867 a(n) = binomial(n,floor(n/2))*(n+1)!.

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A053195 Number of level permutations of degree n.

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A092799 Denominator of partial products in an approximation to Pi/2.

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A193437 Expansion of e.g.f. exp( Sum_{n>=0} x^(3n+1)/(3n+1) ).

A193438 Expansion of e.g.f. exp( Sum_{n>=0} x^(4n+1)/(4n+1) ).