A316728 -id:A316728 - cp's OEIS frontend

A079484 a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882.

1, 3, 45, 1575, 99225, 9823275, 1404728325, 273922023375, 69850115960625, 22561587455281875, 9002073394657468125, 4348001449619557104375, 2500100833531245335015625, 1687568062633590601135546875, 1321365793042101440689133203125

Offset: 0

Benoit Cloitre, Jan 17 2003

nonn

a(n) is the determinant of M(2n+1) where M(k) is the k X k matrix with m(i,j)=j if i+j=k m(i,j)=i otherwise. - Adapted to offset 0, Rainer Rosenthal, Jun 19 2024
In the following two comments on the calculation of the terms using permanents, offset 1 is assumed. In the corresponding PARI code, this is implemented with offset 0. - Hugo Pfoertner, Jun 23 2024
(-1)^n*a(n)/2^(2n-1) is the permanent of the (m X m) matrix {1/(x_i-y_j), 1<=i<=m, 1<=j<=m}, where x_1,x_2,...,x_m are the zeros of x^m-1 and y_1,y_2,...,y_m the zeros of y^m+1 and m=2n-1.
In 1881, R. F. Scott posed a conjecture that the absolute value of permanent of square matrix with elements a(i,j)= (x_i - y_j)^(-1), where x_1,...,x_n are roots of x^n=1, while y_1,...,y_n are roots of y^n=-1, equals a((n-1)/2)/2^n, if n>=1 is odd, and 0, if n>=2 is even. After a century (in 1979), the conjecture was proved by H. Minc. - Vladimir Shevelev, Dec 01 2013
a(n) is the number of permutations in S_{2n+1} in which all cycles have odd length. - José H. Nieto S., Jan 09 2012
Number of 3-bundled increasing bilabeled trees with 2n labels. - Markus Kuba, Nov 18 2014
a(n) is the number of rooted, binary, leaf-labeled topologies with 2n+2 leaves that have n+1 cherry nodes. - Noah A Rosenberg, Feb 12 2019

			G.f. = 1 + 3*x + 45*x^2 + 1575*x^3 + 99225*x^4 + 9823275*x^5 + ...
M(5) =
  [1, 2, 3, 1, 5]
  [1, 2, 2, 4, 5]
  [1, 3, 3, 4, 5]
  [4, 2, 3, 4, 5]
  [1, 2, 3, 4, 5].
Integral_{x=0..oo} x^3*BesselK(1, sqrt(x)) = 1575*Pi. - _Olivier Gérard_, May 20 2009

Miklós Bóna, A walk through combinatorics, World Scientific, 2006.

Alois P. Heinz, Table of n, a(n) for n = 0..224
Cyril Banderier, Markus Kuba, and Michael Wallner, Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions, arXiv:2103.03751 [math.PR], 2021.
Guo-Niu Han and Christian Krattenthaler, Rectangular Scott-type permanents, arXiv:math/0003072 [math.RA], 2000.
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 17 Nov 2014.
MathOverflow, Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?, answer by Tom Copeland posted in Aug 2021.
Henryk Minc, On a conjecture of R. F. Scott (1881), Linear Algebra Appl., Vol. 28 (1979), pp. 141-153.
Theodoros Theodoulidis, On the Closed-Form Expression of Carson’s Integral, Period. Polytech. Elec. Eng. Comp. Sci., Vol. 59, No. 1 (2015), pp. 26-29.
Eric Weisstein's World of Mathematics, Struve function.

Cf. A001818, A000165.
Bisection of A000246, A053195, |A013069|, |A046126|. Cf. A000909.
Cf. A001044, A010791, |A129464|, A114779, are also values of similar moments.
Equals the row sums of A162005.
Cf. A316728.
Diagonal elements of A306364 in even-numbered rows.

I:=[1, 3]; [n le 2 select I[n] else (4*n^2-8*n+3)*Self(n-1): n in [1..20]]; // Vincenzo Librandi, Nov 18 2014

a:= n-> (d-> d(2*n-1)*d(2*n+1))(doublefactorial):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 30 2013
# second Maple program:
A079484 := n-> LinearAlgebra[Determinant](Matrix(2*n+1, (i, j)-> `if`(i+j=2*n+1, j, i))): seq(A079484(n), n=0..14); # Rainer Rosenthal, Jun 18 2024

a[n_] := (2n - 1)!!*(2n + 1)!!; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Jan 30 2013 *)

/* Formula using the zeta function and a log integral:*/
L(n)= intnum(t=0, 1, log(1-1/t)^n);
Zetai(n)= -I*I^n*(2*Pi)^(n-1)/(n-1)*L(1-n);
a(m)={my(n=m+1);round(real(-I*2^(2*n-1)*Zetai(1/2-n)*L(-1/2+n)/(Zetai(-1/2+n)*L(1/2-n))))};
/* Gerry Martens, Mar 07 2011, adapted to offset 0 by Hugo Pfoertner, Jun 19 2024 */

{a(n) = if( n<0, -1 / self()(-1-n), (2*n + 1)! * (2*n)! / (n! * 2^n)^2 )}; /* Michael Somos, May 04 2017 */

{a(n) = if( n<0, -1 / self()(-1-n), my(m = 2*n + 1); m! * polcoeff( x / sqrt( 1 - x^2 + x * O(x^m) ), m))}; /* Michael Somos, May 04 2017 */

\\ using the Pochhammer symbol
a(n) = {my(P(x,k)=gamma(x+k)/gamma(x)); 4^n*round(P(1/2,n)*P(3/2,n))} \\ Hugo Pfoertner, Jun 20 2024

\\ Scott's (1881) method
a(n) = {my(m=2*n+1, X = polroots(x^m-1), Y = polroots(x^m+1), M = matrix(m, m, i, j, 1/(X[i]-Y[j]))); (-1)^n * round(2^m * real(matpermanent(M)))}; \\ Hugo Pfoertner, Jun 23 2024

D-finite with recurrence a(n) = (4*n^2 - 1) * a(n-1) for all n in Z.
a(n) = A001147(n)*A001147(n+1).
E.g.f.: 1/(1-x^2)^(3/2) (with interpolated zeros). - Paul Barry, May 26 2003
a(n) = (2n+1)! * C(2n, n) / 2^(2n). - Ralf Stephan, Mar 22 2004.
Alternatingly signed values have e.g.f. sqrt(1+x^2).
a(n) is the value of the n-th moment of (1/Pi)*BesselK(1, sqrt(x)) on the positive part of the real line. - Olivier Gérard, May 20 2009
a(n) = -2^(2*n-1)*exp(i*n*Pi)*gamma(1/2+n)/gamma(3/2-n). - Gerry Martens, Mar 07 2011
E.g.f. (odd powers) tan(arcsin(x)) = Sum_{n>=0} (2n-1)!!*(2n+1)!!*x^(2*n+1)/(2*n+1)!. - Vladimir Kruchinin, Apr 22 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - ((2*k+2)^2-1)/(1-x/(x - 1/G(k+1))); ( continued fraction ). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = (2^(2*n+3)*Gamma(n+3/2)*Gamma(n+5/2))/Pi. - Jean-François Alcover, Jul 20 2015
Limit_{n->oo} 4^n*(n!)^2/a(n) = Pi/2. - Daniel Suteu, Feb 05 2017
From Michael Somos, May 04 2017: (Start)
a(n) = (2*n + 1) * A001818(n).
E.g.f.: Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)! = x / sqrt(1 - x^2) = tan(arcsin(x)).
Given e.g.f. A(x) = y, then x * y' = y + y^3.
a(n) = -1 / a(-1-n) for all n in Z.
0 = +a(n)*(+288*a(n+2) -60*a(n+3) +a(n+4)) +a(n+1)*(-36*a(n+2) -4*a(n+3)) +a(n+2)*(+3*a(n+2)) for all n in Z. (End)
a(n) = Sum_{k=0..2n} (k+1) * A316728(n,k). - Alois P. Heinz, Jul 12 2018
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + L_1(1)*Pi/2, where L is the modified Struve function.
Sum_{n>=0} (-1)^n/a(n) = 1 - H_1(1)*Pi/2, where H is the Struve function. (End)

Simpler description from Daniel Flath (deflath(AT)yahoo.com), Mar 05 2004

A177042 Eulerian version of the Catalan numbers, a(n) = A008292(2*n+1,n+1)/(n+1).

Original entry on oeis.org

1, 2, 22, 604, 31238, 2620708, 325024572, 55942352184, 12765597850950, 3730771315561300, 1359124435588313876, 603916464771468176392, 321511316149669476991132, 202039976682357297272094824, 147980747895225006590333244088, 124963193751534047864734415925360

Offset: 0

Roger L. Bagula, May 01 2010

nonn

According to the Bidkhori and Sullivant reference's abstract, authors show "that the Eulerian-Catalan numbers enumerate Dyck permutations, [providing] two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analog of the Chung-Feller theorem." - Jonathan Vos Post, Jan 07 2011

Twice the number of permutations of {1,2,...,2n} with n ascents. - Peter Luschny, Jan 11 2011

Alois P. Heinz, Table of n, a(n) for n = 0..220
F. Ardila, The Catalan matroid, J. Combin. Theory Ser. A 104 (2003) 49-62.
Hoda Bidkhori and Seth Sullivant, Eulerian-Catalan Numbers, arXiv:1101.1108 [math.CO], 2011.
Digital Library of Mathematical Functions, Table 26.14.1

Cf. A000108, A008518, A025585, A180056.
Bisection (odd part) of A303287.
Row sums of A316728.

A177042:=func< n | n eq 0 select 1 else 2*(&+[(-1)^k*Binomial(2*n+1,k)*(n-k+1)^(2*n): k in [0..n]]) >;
[A177042(n): n in [0..40]]; // G. C. Greubel, Jun 18 2024

A177042 := proc(n) A008292(2*n+1,n+1)/(n+1) ; end proc:
seq(A177042(n),n=0..10) ; # R. J. Mathar, Jan 08 2011
A177042 := n -> A025585(n+1)/(n+1):
A177042 := n -> `if`(n=0,1,2*A180056(n)):
# The A173018-based recursion below needs no division!
A := proc(n, k) option remember;
       if n = 0 and k = 0 then 1
     elif k > n  or k < 0 then 0
     else (n-k) *A(n-1, k-1) +(k+1) *A(n-1, k)
       fi
     end:
A177042 := n-> `if`(n=0, 1, 2*A(2*n, n)):
seq(A177042(n), n=0..30);
# Peter Luschny, Jan 11 2011

<< DiscreteMath`Combinatorica`
Table[(Eulerian[2*n + 1, n])/(n + 1), {n, 0, 20}]
(* Second program: *)
A[n_, k_] := A[n, k] = Which[n == 0 && k == 0, 1, k > n || k < 0, 0, True, (n - k)*A[n - 1, k - 1] + (k + 1)*A[n - 1, k]]; A177042[n_] := If[n == 0, 1, 2*A[2*n, n]]; Table[A177042[n], {n, 0, 30}] (* Jean-François Alcover, Jul 13 2017, after Peter Luschny *)

def A177042(n): return 2*sum((-1)^k*binomial(2*n+1,k)*(n-k+1)^(2*n) for k in range(n+1)) - int(n==0)
[A177042(n) for n in range(41)] # G. C. Greubel, Jun 18 2024

a(n) = 2*A180056(n) for n > 0, A180056 the central Eulerian numbers in the sense of A173018.
a(n) = A025585(n+1)/(n+1), A025585 the central Eulerian numbers in the sense of A008292.
a(n) = 2 Sum_{k=0..n} (-1)^k binomial(2n+1,k) (n-k+1)^(2n).
a(n) = (n+1)^(-1) Sum_{k=0..n} (-1)^k binomial(2n+2,k)(n+1-k)^(2n+1). - Peter Luschny, Jan 11 2011
a(n) = A008518(2n,n). - Alois P. Heinz, Jun 12 2017
From Alois P. Heinz, Jul 21 2018: (Start)
a(n) = (2n)! * [x^(2n) y^n] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x)).
a(n) = (2n+1)!/(n+1) * [x^(2n+1) y^(n+1)] (1-y)/(1-y*exp((1-y)*x)). (End)

Edited by Alois P. Heinz, Jan 14 2011

A303285 Number of permutations p of [2n] such that the sequence of ascents and descents of p0 forms a Dyck path.

Original entry on oeis.org

1, 1, 8, 172, 7296, 518324, 55717312, 8460090160, 1726791794432, 456440969661508, 151770739970889792, 62022635037246022000, 30564038464166725328768, 17876875858414492985045712, 12245573879235563308351042496, 9711714975145772145881269175104

Offset: 0

Alois P. Heinz, Apr 20 2018

nonn

Here p is a permutation of 1,2,3,...,2n, and p0 refers to the string p followed by 0.
Also the number of permutations p of [2n] such that the sequence of ascents and descents of 0p forms a Dyck path. a(2) = 8: 1432, 2143, 2431, 3142, 3241, 3421, 4132, 4231.
Also the number of permutations p of [2n] that are of odd order and whose M statistic (as defined in the Spiro paper) is equal to n-1. - Sam Spiro, Nov 01 2018

			a(0) = 1: the empty permutation.
a(1) = 1: 12.
a(2) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.

Alois P. Heinz, Table of n, a(n) for n = 0..200
S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv:1810.00993 [math.CO], 2018.
Wikipedia, Counting lattice paths

Bisection (even part) of A303284.
Bisection (even part) of A303287.
Column k=0 of A316728.
Cf. A177042, A180056, A316727, A321280.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(0, 2*n, 0):
seq(a(n), n=0..20);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] + If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[0, 2n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2018, from Maple *)

\\ here b(n) is A177042
b(n)={if(n==0, 1, 2*sum(k=0, n, (-1)^k*binomial(2*n+1,k)*(n-k+1)^(2*n)));}
a(n)={if(n==0, 1, sum(k=1, n, binomial(2*n, 2*k-1)*b(k-1)*b(n-k))/2);} \\ Andrew Howroyd, Nov 01 2018

a(n) ~ c * 2^(2*n) * n^(2*n - 1) / exp(2*n), where c = 8.838022110416151362523442920999767406145711133564692... - Vaclav Kotesovec, May 22 2018
a(n) = (1/2)*Sum_{k odd} binomial(2*n,k)*A177042((k-1)/2)*A177042((2n-k-1)/2) for n>0. - Sam Spiro, Nov 01 2018
a(n) = A321280(2n,n-1) for n >= 1. - Alois P. Heinz, Nov 02 2018

A316727 Number of permutations of {0,1,...,2n} with first element n whose sequence of ascents and descents forms a Dyck path.

Original entry on oeis.org

1, 1, 5, 87, 3337, 223333, 23068057, 3403720071, 679894572497, 176710079709345, 57967294285022281, 23427042148948682599, 11437832700333350250001, 6637473822604173137681381, 4515971399162518697397538173, 3560540787622773257563653593551

Offset: 0

Alois P. Heinz, Jul 11 2018

nonn

All terms are odd.

			a(0) = 1: 0.
a(1) = 1: 120.
a(2) = 5: 23041, 23140, 23410, 24031, 24130.
a(3) = 87: 3401652, 3402165, 3402651, 3405162, ..., 3625041, 3625140, 3645021, 3645120.

Alois P. Heinz, Table of n, a(n) for n = 0..225
Wikipedia, Counting lattice paths

Cf. A303285, A316728.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
    If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] +
    If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[n, n, 0];
a /@ Range[0, 20] (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

a(n) = A316728(n,n).

a(n) ~ c * 4^n * (n!)^2 / n^2, where c = 0.47441051698109564449415497840875665319801746745142596395217012466627... - Vaclav Kotesovec, Jul 15 2018

A316730 Number of permutations of {0,1,...,2n+2} with first element n whose sequence of ascents and descents forms a Dyck path.

Original entry on oeis.org

1, 7, 121, 4411, 283073, 28318137, 4076415425, 798519164779, 204292676593353, 66150225395814649, 26444888796754193841, 12792566645739144488693, 7364969554345555373419625, 4976538708651698959601499559, 3900052284443403730374391636689

Offset: 0

Alois P. Heinz, Jul 11 2018

nonn

All terms are odd.

			a(0) = 1: 021.
a(1) = 7: 12043, 12430, 13042, 13240, 13420, 14032, 14230.
a(2) = 121: 2301654, 2304165, 2304651, 2305164, ..., 2635041, 2635140, 2645031, 2645130.

Alois P. Heinz, Table of n, a(n) for n = 0..224
Wikipedia, Counting lattice paths

Cf. A316728.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(n, n+2, 0):
seq(a(n), n=0..20);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
     If[t > 0,     Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] +
     If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[n, n+2, 0];
a /@ Range[0, 20] (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

a(n) = A316728(n+1,n).

a(n) ~ c * 4^n * (n!)^2, where c = 1.897642067924382577976619913635026612792069869805703855808680498665... - Vaclav Kotesovec, Jul 15 2018

cp's OEIS Frontend

A079484 a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882.

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A177042 Eulerian version of the Catalan numbers, a(n) = A008292(2*n+1,n+1)/(n+1).

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A303285 Number of permutations p of [2n] such that the sequence of ascents and descents of p0 forms a Dyck path.

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A316727 Number of permutations of {0,1,...,2n} with first element n whose sequence of ascents and descents forms a Dyck path.

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A316730 Number of permutations of {0,1,...,2n+2} with first element n whose sequence of ascents and descents forms a Dyck path.

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