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

Author

Simon Plouffe

Keywords

• nonn
• easy

Comments

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

References

• 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).

Links

• 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.

Crossrefs

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

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

Formula

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

Extensions

Edited following a suggestion of Ralf Stephan, Feb 07 2004

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