A336599 -id:A336599 - cp's OEIS frontend

A129890 a(n) = (2n+2)!! - (2n+1)!!.

1, 5, 33, 279, 2895, 35685, 509985, 8294895, 151335135, 3061162125, 68000295825, 1645756410375, 43105900812975, 1214871076343925, 36659590336994625, 1179297174137457375, 40288002704636061375, 1456700757237661060125

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jun 04 2007

Previous name was: Difference between the double factorial of the n-th nonnegative even number and the double factorial of the n-th nonnegative odd number.
In other words, a(n) = b(2n+2)-b(2n+1), where b = A006882. - N. J. A. Sloane, Dec 14 2011 [Corrected Peter Luschny, Dec 01 2014]
a(n) is the number of linear chord diagrams on 2n+2 vertices with one marked chord such that none of the remaining n chords are contained within the marked chord, see [Young]. - Donovan Young, Aug 11 2020

			2!! - 1!! =  2 -  1 =  1;
4!! - 3!! =  8 -  3 =  5;
6!! - 5!! = 48 - 15 = 33.

Selden Crary, Richard Diehl Martinez, and Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
N. Ochiumi, On the total sum of number of nodes covering a given number of leaves in an unordered binary tree
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Cf. A006882, A122649, A202212, A336599.

seq(doublefactorial(2*n+2)-doublefactorial(2*n+1),n=0..9); # Peter Luschny, Dec 01 2014

a[n_] := (2n+2)!! - (2n+1)!!;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 30 2018 *)

E.g.f.: 2/((1-2*x)^2)-1/[(1-2*x)*sqrt(1-2*x)]. - Sergei N. Gladkovskii, Dec 04 2011
a(n) = (2*n+1)*a(n-1) + A000165(n). - Philippe Deléham, Oct 28 2013
Conjecture: a(n) = (2*n + 2)*(2*n + 2)! * Sum_{k >= 1} (-1)^(k+1)/Product_{j = 0..n+1} (k + 2*j). - Peter Bala, Jul 06 2025

New name from Peter Luschny, Dec 01 2014

A035101 E.g.f. x(c(x/2)-1)/(1-2x), where c(x) = g.f. for Catalan numbers A000108.

Original entry on oeis.org

0, 1, 9, 87, 975, 12645, 187425, 3133935, 58437855, 1203216525, 27125492625, 664761133575, 17600023616175, 500706514833525, 15234653491682625, 493699195087473375, 16977671416936605375, 617528830880480644125, 23687738668934964248625

Offset: 1

Wolfdieter Lang

2nd column of triangular array A035342 whose first column is given by A001147(n), n >= 1. Recursion: a(n) = 2*n*a(n-1)+ A001147(n-1), n >= 2, a(1)=0.
a(n) gives the number of organically labeled forests (sets) with two rooted ordered trees with n non-root vertices. See the example a(3)=9 given in A035342. Organic labeling means that the vertex labels along the (unique) path from the root to any of the leaves (degree 1, non-root vertices) is increasing. - Wolfdieter Lang, Aug 07 2007
a(n), n>=2, enumerates unordered n-vertex forests composed of two plane (ordered) ternary (3-ary) trees with increasing vertex labeling. See A001147 (number of increasing ternary trees) and a D. Callan comment there. For a picture of some ternary trees see a W. Lang link under A001764.
a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly 1 of the remaining n-1 chords are contained within the marked chord, see [Young]. - Donovan Young, Aug 11 2020

			a(2)=1 for the forest: {r1-1, r2-2} (with root labels r1 and r2). The order between the components of the forest is irrelevant (like for sets).
a(3)=9 increasing ternary 2-forest with n=3 vertices: there are three 2-forests (the one vertex tree together with any of the three different 2-vertex trees) each with three increasing labelings. - _Wolfdieter Lang_, Sep 14 2007

Robert Israel, Table of n, a(n) for n = 1..370
Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Cf. A000108, A002866, A008549, A336599.

Cf. A001147 (m=1 column of A035342). See a D. Callan comment there on the number of increasing ordered rooted trees on n+1 vertices.

I:=[0,1,9]; [n le 3 select I[n] else - 2*(n-1)*(2*n-3)*Self(n-2)+(4*n-3)*Self(n-1): n in [1..30]]; // Vincenzo Librandi, Sep 12 2015

F:= gfun:-rectoproc({(4*n^2+6*n+2)*a(n)+(-4*n-5)*a(n+1)+a(n+2),a(1)=0,a(2)=1,a(3)=9},a(n),remember):
map(f, [$1..30]); # Robert Israel, Sep 11 2015

Table[Round [n! (4^(n - 1) - Binomial[2 n, n]/2)/2^(n - 1)], {n, 1, 20}] (* Vincenzo Librandi, Sep 12 2015 *)

a(n) = n!*(4^(n-1)-binomial(2*n, n)/2)/2^(n-1);
vector(40, n, a(n)) \\ Altug Alkan, Oct 01 2015

a(n) = n!*A008549(n-1)/2^(n-1) = n!(4^(n-1)-binomial(2*n, n)/2)/2^(n-1).
a(n) = (2n-2)*a(n-1) + A129890(n-2). - Philippe Deléham, Oct 28 2013
a(n) = n!*2^(n-1) - A001147(n) = A002866(n) - A001147(n). - Peter Bala, Sep 11 2015
a(n) = -2*(n-1)*(2*n-3)*a(n-2)+(4*n-3)*a(n-1). - Robert Israel, Sep 11 2015

A336598 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords cross the marked chord.

Original entry on oeis.org

1, 4, 2, 21, 18, 6, 144, 156, 96, 24, 1245, 1500, 1260, 600, 120, 13140, 16470, 16560, 11160, 4320, 720, 164745, 207270, 231210, 194040, 108360, 35280, 5040, 2399040, 2976120, 3507840, 3402000, 2419200, 1149120, 322560, 40320

Offset: 1

Donovan Young, Jul 29 2020

			Triangle begins:
     1;
     4,    2;
    21,   18,    6;
   144,  156,   96,  24;
  1245, 1500, 1260, 600, 120;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can be either (1,3), and so crossed once by (2,4), or (2,4), and so crossed once by (1,3). Hence T(2,1) = 2.

Donovan Young, Table of n, a(n) for n = 1..9870 (Rows 1..140).
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Row sums are n*A001147(n) for n > 0.
First column is A233481(n) for n > 0.
Leading diagonal is A000142(n) for n > 0.
Sub-leading diagonal is n*A000142(n) for n > 1.
Cf. A336599, A336600, A336601.

CoefficientList[Normal[Series[x/Sqrt[1-2*x]/(1-x(1+y)),{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

T(n)={[Vecrev(p) | p<-Vec(serlaplace(x/sqrt(1 - 2*x + O(x^n))/(1 - x*(1 + y))))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jul 29 2020

T(n,k) = n*T(n-1,k) + n*T(n-1,k-1), with T(n,0) = A233481(n) for n > 0.

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

A336600 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords contain the marked chord.

Original entry on oeis.org

1, 5, 1, 32, 11, 2, 260, 116, 38, 6, 2589, 1344, 594, 174, 24, 30669, 17529, 9294, 3774, 984, 120, 422232, 257487, 153852, 76782, 28272, 6600, 720, 6633360, 4234320, 2746260, 1576980, 726480, 242640, 51120, 5040, 117193185, 77358600, 53170380, 33718500, 18171360, 7693200, 2340720, 448560, 40320

Offset: 1

Donovan Young, Jul 30 2020

			Triangle begins:
     1;
     5,    1;
    32,   11,    2;
   260,  116,   38,   6;
  2589, 1344,  594, 174, 24;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can only be (2,3) and it is contained by one other chord, namely (1,4), hence T(2,1) = 1.

Donovan Young, Table of n, a(n) for n = 1..9870
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Row sums are n*A001147(n) for n > 0.
Leading diagonal is A000142(n-1) for n > 0.
Sub-leading diagonal is A001344(n-2) for n > 1.
Cf. A336598, A336599, A336601.

CoefficientList[Normal[Series[Log[(1-x*(1+y))/(1-2*x)]/(1-y)/Sqrt[1-2*x],{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

E.g.f.: log((1 - x*(1 + y))/(1 - 2*x))/(1 - y)/sqrt(1 - 2*x).

A336601 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are excluded by (i.e., are outside and do not contain) the marked chord.

Original entry on oeis.org

1, 4, 2, 22, 16, 7, 160, 136, 88, 36, 1464, 1344, 1044, 624, 249, 16224, 15504, 13344, 9624, 5484, 2190, 211632, 206592, 188952, 152832, 104322, 58080, 23535, 3179520, 3139200, 2977920, 2594880, 1990080, 1309680, 725040, 299880, 54092160, 53729280, 52096320, 47681280, 39652560, 29174400, 18809640, 10473120, 4426065

Offset: 1

Donovan Young, Jul 31 2020

			Triangle begins:
     1;
     4,    2;
    22,   16,    7;
   160,  136,   88,  36;
  1464, 1344, 1044, 624, 249;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can either be (1,2) and it excludes one other chord, namely (3,4), or vice-versa, hence T(2,1) = 2.

Donovan Young, Table of n, a(n) for n = 1..9870
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Row sums are n*A001147(n) for n > 0.
The first column is A087547(n) for n > 0.
Leading diagonal is A034430(n-1) for n > 0.
Cf. A336598, A336599, A336600.

CoefficientList[Normal[Series[1/(1-y)/Sqrt[1-2*x]*ArcTan[(x*(1-y))/Sqrt[(1-2*x)]/Sqrt[1-2*y*x]],{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

E.g.f.: arctan(x*(1 - y)/sqrt((1 - 2*x)*(1 - 2*x*y)))/(1 - y)/sqrt(1 - 2*x).

cp's OEIS Frontend

A129890 a(n) = (2*n+2)!! - (2*n+1)!!.

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A035101 E.g.f. x*(c(x/2)-1)/(1-2*x), where c(x) = g.f. for Catalan numbers A000108.

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A336598 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords cross the marked chord.

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A336600 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords contain the marked chord.

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A336601 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are excluded by (i.e., are outside and do not contain) the marked chord.

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Mathematica

Formula

A129890 a(n) = (2n+2)!! - (2n+1)!!.

A035101 E.g.f. x(c(x/2)-1)/(1-2x), where c(x) = g.f. for Catalan numbers A000108.