A193651 -id:A193651 - cp's OEIS frontend

A001879 a(n) = (2n+2)!/(n!*2^(n+1)).

1, 6, 45, 420, 4725, 62370, 945945, 16216200, 310134825, 6547290750, 151242416325, 3794809718700, 102776096548125, 2988412653476250, 92854250304440625, 3070380543400170000, 107655217802968460625, 3989575718580595893750, 155815096120119939628125

Offset: 0

N. J. A. Sloane

From Wolfdieter Lang, Oct 06 2008: (Start)
a(n) is the denominator of the n-th approximant to the continued fraction 1^2/(6+3^2/(6+5^2/(6+... for Pi-3. W. Lang, Oct 06 2008, after an e-mail from R. Rosenthal. Cf. A142970 for the corresponding numerators.
The e.g.f. g(x)=(1+x)/(1-2*x)^(5/2) satisfies (1-4*x^2)*g''(x) - 2*(8*x+3)*g'(x) -9*g(x) = 0 (from the three term recurrence given below). Also g(x)=hypergeom([2,3/2],[1],2*x). (End)
Number of descents in all fixed-point-free involutions of {1,2,...,2(n+1)}. A descent of a permutation p is a position i such that p(i) > p(i+1). Example: a(1)=6 because the fixed-point-free involutions 2143, 3412, and 4321 have 2, 1, and 3 descents, respectively. - Emeric Deutsch, Jun 05 2009
First differences of A193651. - Vladimir Reshetnikov, Apr 25 2016
a(n-2) is the number of maximal elements in the absolute order of the Coxeter group of type D_n. - Jose Bastidas, Nov 01 2021

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77 (Problem 10, values of Bessel polynomials).
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 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 1.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
J. Riordan, Notes to N. J. A. Sloane, Jul. 1968

Cf. A002544, A001814, A001876, A001877, A001878.
Second column of triangle A001497. Equals (A001147(n+1)-A001147(n))/2.
Equals row sums of A163938.

[Factorial(2*n+2)/(Factorial(n)*2^(n+1)): n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

restart: G(x):=(1-x)/(1-2*x)^(1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=2..20); # Zerinvary Lajos, Apr 04 2009

Table[(2n+2)!/(n!2^(n+1)),{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)

PARI
```
a(n)=if(n<0,0,(2*n+2)!/n!/2^(n+1))
```

E.g.f.: (1+x)/(1-2*x)^(5/2).
a(n)*n = a(n-1)*(2n+1)*(n+1); a(n) = a(n-1)*(2n+4)-a(n-2)*(2n-1), if n>0. - Michael Somos, Feb 25 2004
From Wolfdieter Lang, Oct 06 2008: (Start)
a(n) = (n+1)*(2*n+1)!! with the double factorials (2*n+1)!!=A001147(n+1).
D-finite with recurrence a(n) = 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(-1)=0, a(0)=1. (End)
With interpolated 0's, e.g.f.: B(A(x)) where B(x)= x exp(x) and A(x)=x^2/2.
E.g.f.: -G(0)/2 where G(k) = 1 - (2*k+3)/(1 - x/(x - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
G.f.: (1-x)/(2*x^2*Q(0)) - 1/(2*x^2), where Q(k) = 1 - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
From Karol A. Penson, Jul 12 2013: (Start)
Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),
  w(x) = -(1/4)*sqrt(2)*sqrt(x)*(1-x)*exp(-x/2)/sqrt(Pi):
  a(n) = Integral_{x>=0} x^n*w(x), n>=0.
  For x>1, w(x)>0. w(0)=w(1)=limit(w(x),x=infinity)=0. For x<1, w(x)<0.
Asymptotics: a(n)->(1/576)*2^(1/2+n)*(1152*n^2+1680*n+505)*exp(-n)*(n)^(n), for n->infinity. (End)
G.f.: 2F0(3/2,2;;2x). - R. J. Mathar, Aug 08 2015

Entry revised Aug 31 2004 (thanks to Ralf Stephan and Michael Somos)

E.g.f. in comment line corrected by Wolfdieter Lang, Nov 21 2011

A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

Original entry on oeis.org

1, 1, 3, 5, 15, 33, 91, 221, 583, 1465, 3795, 9653, 24831, 63441, 162763, 416525, 1067575, 2733673, 7003971, 17938661, 45954543, 117709185, 301527355, 772364093, 1978473511

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

Suppose that p=p(0)*x^n+p(1)*x^(n-1)+...+p(n-1)*x+p(n) is a polynomial of positive degree and that Q is a sequence of polynomials: q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k), for k=0,1,2,... The Q-downstep of p is the polynomial given by D(p)=p(0)*q(n-1,x)+p(1)*q(n-2,x)+...+p(n-1)*q(0,x)+p(n).

Since degree(D(p))

Example: let p(x)=2*x^3+3*x^2+4*x+5 and q(k,x)=(x+1)^k.

D(p)=2(x+1)^2+3(x+1)+4(1)+5=2x^2+7x+14

D(D(p))=2(x+1)+7(1)+14=2x+23

D(D(D(p)))=2(1)+23=25;

the Q-residue of p is 25.

We may regard the sequence Q of polynomials as the triangular array formed by coefficients:

t(0,0)

t(1,0)....t(1,1)

t(2,0)....t(2,1)....t(2,2)

t(3,0)....t(3,1)....t(3,2)....t(3,3)

and regard p as the vector (p(0),p(1),...,p(n)). If P is a sequence of polynomials [or triangular array having (row n)=(p(0),p(1),...,p(n))], then the Q-residues of the polynomials form a numerical sequence.

Following are examples in which Q is the triangle given by t(i,j)=1 for 0<=i<=j:

Q.....P...................Q-residue of P

1.....1...................A000079, 2^n

1....(x+1)^n..............A007051, (1+3^n)/2

1....(x+2)^n..............A034478, (1+5^n)/2

1....(x+3)^n..............A034494, (1+7^n)/2

1....(2x+1)^n.............A007582

1....(3x+1)^n.............A081186

1....(2x+3)^n.............A081342

1....(3x+2)^n.............A081336

1.....A040310.............A193649

1....(x+1)^n+(x-1)^n)/2...A122983

1....(x+2)(x+1)^(n-1).....A057198

1....(1,2,3,4,...,n)......A002064

1....(1,1,2,3,4,...,n)....A048495

1....(n,n+1,...,2n).......A087323

1....(n+1,n+2,...,2n+1)...A099035

1....p(n,k)=(2^(n-k))*3^k.A085350

1....p(n,k)=(3^(n-k))*2^k.A090040

1....A008288 (Delannoy)...A193653

1....A054142..............A101265

1....cyclotomic...........A193650

1....(x+1)(x+2)...(x+n)...A193651

1....A114525..............A193662

More examples:

Q...........P.............Q-residue of P

(x+1)^n...(x+1)^n.........A000110, Bell numbers

(x+1)^n...(x+2)^n.........A126390

(x+2)^n...(x+1)^n.........A028361

(x+2)^n...(x+2)^n.........A126443

(x+1)^n.....1.............A005001

(x+2)^n.....1.............A193660

A094727.....1.............A193657

(k+1).....(k+1)...........A001906 (even-ind. Fib. nos.)

(k+1).....(x+1)^n.........A112091

(x+1)^n...(k+1)...........A029761

(k+1)......A049310........A193663

(In these last four, (k+1) represents the triangle t(n,k)=k+1, 0<=k<=n.)

A051162...(x+1)^n.........A193658

A094727...(x+1)^n.........A193659

A049310...(x+1)^n.........A193664

A075362....A075362........A193665

Changing the notation slightly leads to the Mathematica program below and the following formulation for the Q-downstep of p: first, write t(n,k) as q(n,k). Define r(k)=Sum{q(k-1,i)*r(k-1-i) : i=0,1,...,k-1} Then row n of D(p) is given by v(n)=Sum{p(n,k)*r(n-k) : k=0,1,...,n}.

			First five rows of Q, coefficients of Fibonacci polynomials (A049310):
1
1...0
1...0...1
1...0...2...0
1...0...3...0...1
To obtain a(4)=15, downstep four times:
D(x^4+3*x^2+1)=(x^3+x^2+x+1)+3(x+1)+1: (1,1,4,5) [coefficients]
DD(x^4+3*x^2+1)=D(1,1,4,5)=(1,2,11)
DDD(x^4+3*x^2+1)=D(1,2,11)=(1,14)
DDDD(x^4+3*x^2+1)=D(1,14)=15.

Cf. A192872 (polynomial reduction), A193091 (polynomial augmentation), A193722 (the upstep operation and fusion of polynomial sequences or triangular arrays).

q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
f[n_, x_] := Fibonacci[n + 1, x];
p[n_, k_] := Coefficient[f[n, x], x, k]; (* A049310 *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 24}]    (* A193649 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]

Conjecture: G.f.: -(1+x)*(2*x-1) / ( (x-1)*(4*x^2+x-1) ). - R. J. Mathar, Feb 19 2015

A275527 Number of distinct classes of permutations of length n under reversal and complement to n+1.

Original entry on oeis.org

1, 1, 1, 4, 12, 64, 360, 2544, 20160, 181632

Offset: 1

Olivier Gérard, Jul 31 2016

Let us consider two permutations to be equivalent if they can be obtained from each other by cyclic rotation (12345->(23451,34512,45123,51234) or n+1-complement (31254->35412), or a combination of those two transformations (they commute with each other). a(n) is the number of classes.
We obtain the same number of classes if the transformations are (addition of a constant modulo n and reversal (12345->54321)) but not the same set of representatives.
It seems probable that a(2n+1) = (2n)!/2
This sequence may be related to A113247 (and A113248) as they share a common dissection 1, 4, 64, 2544, 181632. The fact that they count permutation classes for the major index is a further indication.
Number of path necklaces, defined as equivalence classes of (labeled, undirected) Hamiltonian paths under rotation of the vertices. The cycle version is A000939. - Gus Wiseman, Mar 02 2019

			Examples of permutation representatives. The representative is chosen to be the first of the class in lexicographic order.
n=4 case addition mod n and reversal
1234, 1243, 1324, 1423.
n=4 case rotation and complement
1234, 1243, 1324, 1342.
.
n=5 case addition mod n and reversal
12345, 12354, 12435, 12453, 12534, 13245, 13425, 13452, 13524, 14235, 14523, 15234.
n=5 case rotation and complement
12345, 12354, 12435, 12453, 12534, 13245, 13425, 13452, 13524, 14235, 14325, 14352.

Wikipedia, Hamiltonian path.
Wikipedia, Necklace (combinatorics).
Gus Wiseman, Inequivalent representatives of the a(5) = 12 path necklaces.
Gus Wiseman, Inequivalent representatives of the a(6) = 54 path necklaces.

Cf. A000939, A000940, A002619, A089066, A262480 (other symmetry classes of permutations).
Cf. A193651 (inspiration for a(2n)).
Cf. A000031, A006125, A008965, A059966, A060223, A192332, A323858, A323870, A324461.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#,2,1]]&/@Permutations[Range[n]]],#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&]],{n,8}] (* Gus Wiseman, Mar 02 2019 *)

(Conjecture). If n odd a(n)=((n - 1))!/2. If n even a(n)= 1/2 (n - 2)!! (1 + ( n - 1)!!).

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A001879 a(n) = (2n+2)!/(n!*2^(n+1)).

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A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

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A275527 Number of distinct classes of permutations of length n under reversal and complement to n+1.

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