A007228 -id:A007228 - cp's OEIS frontend

A007226 a(n) = 2*det(M(n; -1))/det(M(n; 0)), where M(n; m) is the n X n matrix with (i,j)-th element equal to 1/binomial(n + i + j + m, n).

2, 3, 10, 42, 198, 1001, 5304, 29070, 163438, 937365, 5462730, 32256120, 192565800, 1160346492, 7048030544, 43108428198, 265276342782, 1641229898525, 10202773534590, 63698396932170, 399223286267190, 2510857763851185, 15842014607109600

Offset: 0

Simon Plouffe

For n >= 1, a(n) is the number of distinct perforation patterns for deriving (v,b) = (n+1,n) punctured convolutional codes from (3,1). [Edited by Petros Hadjicostas, Jul 27 2020]
Apparently Bégin's (1992) paper was presented at a poster session at the conference and was never published.
a(n) is the total number of down steps between the first and second up steps in all 2-Dyck paths of length 3*(n+1). A 2-Dyck path is a nonnegative lattice path with steps (1,2), (1,-1) that starts and ends at y = 0. - Sarah Selkirk, May 07 2020
From Petros Hadjicostas, Jul 27 2020: (Start)
"A punctured convolutional code is a high-rate code obtained by the periodic elimination (i.e., puncturing) of specific code symbols from the output of a low-rate encoder. The resulting high-rate code depends on both the low-rate code, called the original code, and the number and specific positions of the punctured symbols." (The quote is from Haccoun and Bégin (1989).)
A high-rate code (v,b) (written as R = b/v) can be constructed from a low-rate code (v0,1) (written as R = 1/v0) by deleting from every v0*b code symbols a number of v0*b - v symbols (so that the resulting rate is R = b/v).
Even though my formulas below do not appear in the two published papers in the IEEE Transactions on Communications, from the theory in those two papers, it makes sense to replace "k|b" with "k|v0*b" (and "k|gcd(v,b)" with "k|gcd(v,v0*b)"). Pab Ter, however, uses "k|b" in the Maple programs in the related sequences A007223, A007224, A007225, A007227, and A007229. (End)
Conjecture: a(n) is odd iff n = A022341(k) for some k. - Peter Bala, Mar 13 2023

Guy Bégin, On the enumeration of perforation patterns for punctured convolutional codes, Séries Formelles et Combinatoire Algébrique, 4th colloquium, 15-19 Juin 1992, Montréal, Université du Québec à Montréal, pp. 1-10.
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..198
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Guy Bégin and David Haccoun, High rate punctured convolutions codes: Structure properties and construction techniques, IEEE Transactions on Communications 37(12) (1989), 1381-1385.
Ryan C. Chen, Yujin H. Kim, Jared D. Lichtman, Steven J. Miller, Shannon Sweitzer, and Eric Winsor, Spectral Statistics of Non-Hermitian Random Matrix Ensembles, arXiv:1803.08127 [math-ph], 2018.
Fabio Deelan Cunden, Marilena Ligabò, and Tommaso Monni, Random matrices associated to Young diagrams, arXiv:2301.13555 [math.PR], 2023. See p. 7.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv:1410.5747 [math.CO], 2014.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, 339(3) (2016), 1116-1139.
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
N. S. S. Gu, H. Prodinger, and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat. 31 (2010), 720-732, Theorem 2 at k = 2.
David Haccoun and Guy Bégin, High rate punctured convolutional codes for Viterbi and sequential coding, IEEE Transactions on Communications, 37(11) (1989), 1113-1125; see Section II.
W. Mlotkowski and K. A. Penson, The probability measure corresponding to 2-plane trees, arXiv:1304.6544 [math.PR], 2013.
Jocelyn Quaintance, Combinatoric Enumeration of Two-Dimensional Proper Arrays, Discrete Math., 307 (2007), 1844-1864.

Cf. A006013, A007223, A007224, A007225, A007227, A007228, A007229, A124724, A000139.

[Binomial(3*n,n)/(2*n+1)+Binomial(3*n+1,n)/(n+1): n in [0..25]]; // Vincenzo Librandi, Aug 10 2014

A007226:=n->2*binomial(3*n,n)-binomial(3*n,n+1): seq(A007226(n), n=0..30); # Wesley Ivan Hurt, Aug 11 2014

Table[2*Binomial[3n,n]-Binomial[3n,n+1], {n,0,20}] (* Harvey P. Dale, Aug 10 2014 *)

a(n) = {my(M1=matrix(n,n)); my(M0=matrix(n,n)); for(i=1, n, for(j=1, n, M1[i,j] = 1/binomial(n+i+j-1,n); M0[i,j] = 1/binomial(n+i+j,n);)); 2*matdet(M1)/matdet(M0);} \\ Petros Hadjicostas, Jul 27 2020

a(n) = (2/(n + 1))*binomial(3*n, n).
a(n) = (2n+1) * A000139(n). - F. Chapoton, Feb 23 2024
a(n) = 2*C(3*n, n) - C(3*n, n+1) for n >= 0. - David Callan, Oct 25 2004
a(n) = C(3*n, n)/(2*n + 1) + C(3*n + 1, n)/(n + 1) = C(3*n, n)/(2*n + 1) + 2*C(3*n + 1, n)/(2*n + 2) for n >= 0. - Paul Barry, Nov 05 2006
G.f.: g*(2 - g)/x, where g*(1 - g)^2 = x. - Mark van Hoeij, Nov 08 2011 [Thus, g = (4/3)*sin((1/3)*arcsin(sqrt(27*x/4)))^2. - Petros Hadjicostas, Jul 27 2020]
Recurrence: 2*(n+1)*(2*n-1)*a(n) - 3*(3*n-1)*(3*n-2)*a(n-1) = 0 for n >= 1. - R. J. Mathar, Nov 26 2012
G.f.: (1 - 1/B(x))/x, where B(x) is the g.f. of A006013. [Vladimir Kruchinin, Mar 05 2013]
G.f.: ( -16 * sin(asin((3^(3/2) * sqrt(x))/2)/3)^4 + 24 * sin(asin((3^(3/2) * sqrt(x))/2)/3)^2 ) / (9*x). [Vladimir Kruchinin, Nov 16 2013]
From Petros Hadjicostas, Jul 27 2020: (Start)
The number of perforation patterns to derive high-rate convolutional code (v,b) (written as R = b/v) from a given low-rate convolutional code (v0, 1) (written as R = 1/v0) is (1/b)*Sum_{k|gcd(v,b)} phi(k)*binomial(v0*b/k, v/k).
According to Pab Ter's Maple code in the related sequences (see above), this is the coefficient of z^v in the polynomial (1/b)*Sum_{k|b} phi(k)*(1 + z^k)^(v0*b/k).
Here (v,b) = (n+1,n) and (v0,1) = (3,1), so for n >= 1,
a(n) = (1/n)*Sum_{k|gcd(n+1,n)} phi(k)*binomial(3*n/k, (n+1)/k).
This simplifies to
a(n) = (1/n)*binomial(3*n, n+1) for n >= 1. (End)
G.f.: (-(p+r)*(4*r-12*p)^(1/3)+(p-r)*(4*r+12*p)^(1/3)+8)/(12*z), where p = i*sqrt(3*z), r = sqrt(4-27*z), and i = sqrt(-1) is the imaginary unit. - Karol A. Penson, Mar 20 2025

Edited following a suggestion of Ralf Stephan, Feb 07 2004

Offset changed to 0 and all formulas checked by Petros Hadjicostas, Jul 27 2020

A124724 a(n) = (4/(n + 1)) * C(5*n, n).

Original entry on oeis.org

4, 10, 60, 455, 3876, 35420, 339300, 3362260, 34179860, 354465254, 3735373880, 39884521950, 430571952300, 4691735290080, 51534335175776, 570003171679020, 6343110854237300, 70968228417131850, 797820661622862900

Offset: 0

Paul Barry, Nov 05 2006

a(n) is the total number of down steps between the first and second up steps in all 4-Dyck paths of length 5*(n+1). A 4-Dyck path is a nonnegative lattice path with steps (1,4), (1,-1) that starts and ends at y = 0. - Sarah Selkirk, May 07 2020

Michael De Vlieger, Table of n, a(n) for n = 0..924
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.
Fabio Deelan Cunden, Marilena Ligabò, and Tommaso Monni, Random matrices associated to Young diagrams, arXiv:2301.13555 [math.PR], 2023. See p. 7.
N. S. S. Gu, H. Prodinger, and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat. 31 (2010) 720-732; see Theorem 2 with k = 4.

Cf. A007226, A007228.

Array[(4/(# + 1))*Binomial[5 #, #] &, 28, 0] (* Michael De Vlieger, Apr 12 2023 *)

a(n) = (4/(n+1)) * binomial(5*n,n); \\ Michel Marcus, May 08 2020

a(n) = C(5*n, n)/(4*n + 1) + 2*C(5*n + 1, n)/(4*n + 2) + 3*C(5*n + 2, n)/(4*n + 3) + 4*C(5*n + 3, n)/(4*n + 4).

A334642 a(n) is the total number of down steps between the first and second up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

Original entry on oeis.org

0, 3, 9, 32, 139, 669, 3430, 18360, 101403, 573551, 3305445, 19340100, 114579348, 685962172, 4143459504, 25220816752, 154545611355, 952583230899, 5902090839715, 36738469359480, 229636903762035, 1440759023752125, 9070230371741490, 57278432955350880

Offset: 0

Benjamin Hackl, May 07 2020

For n = 1, there is no 2nd up step, a(1) = 3 enumerates the total number of down steps between the 1st up step and the end of the path.

			For n = 1, the 2_1-Dyck paths are UDD, DUD. This corresponds to a(1) = 2 + 1 = 3 down steps between the 1st up step and the end of the path.
For n = 2, the 2_1-Dyck paths are UUDDDD, UDUDDD, UDDUDD, UDDDUD, DUDDUD, DUDUDD, DUUDDD. In total, there are a(2) = 0 + 1 + 2 + 3 + 2 + 1 + 0 = 9 down steps between the 1st and 2nd up step.

A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A007226, A007228, A124724.

a[0] = 0; a[n_] := 2 * Binomial[3*n, n]/(n + 1) - Binomial[3*n + 1, n]/(n + 1) + 4 * Binomial[3*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 24, 0] (* Amiram Eldar, May 09 2020 *)

a(n) = if (n==0, 0, 2*binomial(3*n, n)/(n+1) - binomial(3*n+1, n)/(n+1) + 4*binomial(3*(n-1), n-1)/n - 2*(n==1)); \\ Michel Marcus, May 09 2020

a(0) = 0 and a(n) = 2*binomial(3*n, n)/(n+1) - binomial(3*n+1, n)/(n+1) + 4*binomial(3*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket.

A334645 a(n) is the total number of down steps between the 2nd and 3rd up steps in all 3-Dyck paths of length 4*n. A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.

Original entry on oeis.org

0, 0, 18, 52, 277, 1752, 12120, 88692, 674751, 5282160, 42267384, 344152080, 2842055359, 23746693240, 200383750632, 1705243729560, 14617677294675, 126106202849760, 1094034474058488, 9538676631305712, 83536778390997780, 734521734171474400, 6481894477750488160

Offset: 0

Benjamin Hackl, May 12 2020

For n = 2, there is no 3rd up step, a(2) = 18 enumerates the total number of down steps between the 2nd up step and the end of the path.

			For n = 2, the 3-Dyck paths are UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 3 + 4 + 5 + 6 = 18 down steps between the 2nd up step and the end of the path.

A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334640, A334643, A334682, A334785.

[3*sum([binomial(4*j + 1, j)*binomial(4*(n - j), n - j)/(4*j + 1)/(n - j + 1) for j in srange(1, 3)]) if n > 1 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = 0 and a(n) = 3*Sum_{j=0..2} binomial(4*j+1, j) * binomial(4*(n-j), n-j)/((4*j+1) * (n-j+1)) for n > 1.

A334647 a(n) is the total number of down steps between the first and second up steps in all 3_1-Dyck paths of length 4*n.

Original entry on oeis.org

0, 5, 16, 78, 470, 3153, 22588, 169188, 1308762, 10374460, 83829856, 687929086, 5717602930, 48030047206, 407142435000, 3478286028840, 29917720938690, 258866494630164, 2251694583485824, 19677972159742360, 172694287830500440, 1521328368800877065

Offset: 0

Benjamin Hackl, May 12 2020

A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

For n = 1, there is no 2nd up step, a(1) = 5 enumerates the total number of down steps between the 1st up step and the end of the path.

			For n = 1, the 3_1-Dyck paths are UDDD, DUDD. This corresponds to a(1) = 3 + 2 = 5 down steps between the 1st up step and the end of the path.
For n = 2, the 3_1-Dyck paths are DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 3 + 2 + 1 + 0 + 4 + 3 + 2 + 1 + 0 = 16 down steps between the 1st and 2nd up step.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334645, A334646, A334648, A334649, A334680, A334682, A334785.

a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n + 1) - 2 * Binomial[4*n + 1, n]/(n + 1) + 6 * Binomial[4*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)

[3*binomial(4*n, n)/(n+1) - 2*binomial(4*n+1, n)/(n+1) + 6*binomial(4*(n-1), n-1)/n - 2*(n==1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - 2*binomial(4*n+1, n)/(n+1) + 6*binomial(4*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket.

A334640 a(n) is the total number of down steps between the 2nd and 3rd up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.

Original entry on oeis.org

0, 0, 9, 19, 72, 324, 1595, 8307, 44982, 250648, 1427679, 8274825, 48644310, 289334160, 1738043892, 10529070020, 64252519830, 394601627376, 2437058926871, 15126463230165, 94306717535940, 590318477063700, 3708527622652755, 23374587898663155, 147770791807427880

Offset: 0

Benjamin Hackl, May 07 2020

For n = 2, there is no 3rd up step, a(2) = 9 enumerates the total number of down steps between the 2nd up step and the end of the path.

			For n = 2, there are the 2-Dyck paths UUDDDD, UDUDDD, UDDUDD. Between the 2nd up step and the end of the path there are a(2) = 4 + 3 + 2 = 9 down steps in total.

Alois P. Heinz, Table of n, a(n) for n = 0..1212
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A007226, A007228, A124724, A334641, A334642, A334682.

b:= proc(x, y, u, c) option remember; `if`(x=0, c,
     `if`(y+20, b(x-1, y-1, u, c+`if`(u=2, 1, 0)), 0))
    end:
a:= n-> b(3*n, 0$3):
seq(a(n), n=0..24);  # Alois P. Heinz, May 09 2020
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0$2, 9][n+1],
     (3*(n-1)*(3*n-8)*(3*n-7)*(13*n-20)*a(n-1))/
     (2*(13*n-33)*(n-2)*(2*n-3)*n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, May 09 2020

a[0] = a[1] = 0; a[n_] := 2 * Sum[Binomial[3*j + 1, j] * Binomial[3*(n - j), n - j]/((3*j + 1)*(n - j + 1)), {j, 1, 2}]; Array[a, 25, 0] (* Amiram Eldar, May 09 2020 *)

a(n) = if (n<=1, 0, 2*sum(j=1, 2, binomial(3*j+1,j) * binomial(3*(n-j),n-j)/((3*j+1)*(n-j+1)))); \\ Michel Marcus, May 09 2020

a(0) = a(1) = 0 and a(n) = 2*Sum_{j=1..2} binomial(3*j+1,j) * binomial(3*(n-j),n-j) / ((3*j+1)*(n-j+1)) for n > 1.

A334646 a(n) is the total number of down steps between the 3rd and 4th up steps in all 3-Dyck paths of length 4*n.

Original entry on oeis.org

0, 0, 0, 118, 409, 2368, 15750, 112716, 845295, 6551208, 52035714, 421286280, 3463401007, 28832656408, 242565115858, 2058945519936, 17611312647075, 151647023490480, 1313460091978458, 11435310622320552, 100019000856225156, 878443730199290560

Offset: 0

Benjamin Hackl, May 12 2020

A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.

			For n = 3, there is no 4th up step, a(3) = 118 enumerates the total number of down steps between the 3rd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334641, A334645, A334682, A334785.

a[0] = a[1] = a[2] = 0; a[n_] := 3 * Sum[Binomial[4*j + 1, j] * Binomial[4*(n - j), n - j]/((4*j + 1)*(n - j + 1)), {j, 1, 3}]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)

[3*sum([binomial(4*j + 1, j)*binomial(4*(n - j), n - j)/(4*j + 1)/(n - j + 1) for j in srange(1, 4)]) if n > 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = a(2) = 0 and a(n) = 3*Sum_{j=1..3} binomial(4*j+1, j)*binomial(4*(n-j), n-j)/((4*j+1)*(n-j+1)) for n > 2.

A334648 a(n) is the total number of down steps between the second and third up steps in all 3_1-Dyck paths of length 4*n.

Original entry on oeis.org

0, 0, 34, 132, 722, 4638, 32416, 238956, 1827918, 14370595, 115384756, 942115942, 7798224226, 65286060253, 551838621972, 4702955036640, 40366238473530, 348631520142879, 3027590307082804, 26420699531880832, 231571468023697960, 2037650653547067005

Offset: 0

Benjamin Hackl, May 12 2020

A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

For n = 2, there is no 3rd up step, a(2) = 34 enumerates the total number of down steps between the 2nd up step and the end of the path.

			For n = 2, the 3_1-Dyck paths are DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 34 down steps between the 2nd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334649, A334680, A334682, A334785.

a[0] = a[1] = 0; a[n_] := Binomial[4*n + 1, n]/(4*n + 1) + 6 * Sum[Binomial[4*j + 2, j] * Binomial[4*(n - j), n - j]/((4*j + 2)*(n - j + 1)), {j, 1, 2}] - 9 * Boole[n == 2]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)

[binomial(4*n + 1, n)/(4*n + 1) + 6*sum([binomial(4*j + 2, j)*binomial(4*(n - j), n - j)/(4*j + 2)/(n - j + 1) for j in srange(1, 3)]) - 9*(n==2) if n > 1 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = 0 and a(n) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..2} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 9*[n=2] for n > 1, where [ ] is the Iverson bracket.

A334649 a(n) is the total number of down steps between the third and fourth up steps in all 3_1-Dyck paths of length 4*n.

Original entry on oeis.org

0, 0, 0, 236, 1034, 6094, 40996, 295740, 2231022, 17370163, 138473536, 1124433142, 9266859394, 77307427741, 651540030688, 5538977450256, 47442103851930, 409000732566399, 3546232676711824, 30903652601552272, 270529448396053576, 2377829916885541565

Offset: 0

Benjamin Hackl, May 12 2020

A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

For n = 3, there is no 4th up step, a(3) = 236 enumerates the total number of down steps between the 3rd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334648, A334680, A334682, A334785.

[binomial(4*n + 1, n)/(4*n + 1) + 6*sum([binomial(4*j + 2, j)*binomial(4*(n - j), n - j)/(4*j + 2)/(n - j + 1) for j in srange(1, 4)]) - 52*(n==3) if n > 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = a(2) = 0 and a(n) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..3} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 52*[n=3] for n > 2, where [ ] is the Iverson bracket.

cp's OEIS Frontend

A024497 Duplicate of A007228.

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A007226 a(n) = 2*det(M(n; -1))/det(M(n; 0)), where M(n; m) is the n X n matrix with (i,j)-th element equal to 1/binomial(n + i + j + m, n).

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A124724 a(n) = (4/(n + 1)) * C(5*n, n).

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A334642 a(n) is the total number of down steps between the first and second up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

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PARI

Formula

A334645 a(n) is the total number of down steps between the 2nd and 3rd up steps in all 3-Dyck paths of length 4*n. A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.

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SageMath

Formula

A334647 a(n) is the total number of down steps between the first and second up steps in all 3_1-Dyck paths of length 4*n.

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Mathematica

SageMath

Formula

A334640 a(n) is the total number of down steps between the 2nd and 3rd up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.

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Maple

Mathematica

PARI

Formula

A334646 a(n) is the total number of down steps between the 3rd and 4th up steps in all 3-Dyck paths of length 4*n.