A007228 a(n) = 3*binomial(4*n,n)/(n+1). (Formerly M5200)

3, 6, 28, 165, 1092, 7752, 57684, 444015, 3506100, 28242984, 231180144, 1917334783, 16077354108, 136074334200, 1160946392760, 9973891723635, 86210635955220, 749191930237608, 6541908910355280, 57369142749576660, 505045163173167760, 4461713825057817120

Offset

0,1

Comments

For n >= 1, a(n) is the number of distinct perforation patterns for deriving (v,b) = (n+1,n) punctured convolutional codes from (4,1). [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 3-Dyck paths of length 4*(n+1). A 3-Dyck path is a nonnegative lattice path with steps (1,3), (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: for n >= 1, a(n) is odd iff n = 4*A263133(k) + 3 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

Andrew Howroyd, Table of n, a(n) for n = 0..500

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

Guy Bégin and David Haccoun, High rate punctured convolutions codes: Structure properties and construction techniques, IEEE Transactions on Communications 37(12) (1989), 1381-1385.

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, Theorem 2 at k=3.

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.

Formula

a(n) = C(4*n,n)/(3*n+1) + 2*C(4*n+1,n)/(3*n+2) + 3*C(4*n+2,n)/(3*n+3). - Paul Barry, Nov 05 2006

G.f.: g + g^2 + g^3 where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011

3*(3*n-1)*(3*n-2)*(n+1)*a(n) - 8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012

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) = (4,1), so for n >= 1,

a(n) = (1/n)*Sum_{k|gcd(n+1,n)} phi(k)*binomial(4*n/k, (n+1)/k).

This simplifies to

a(n) = (1/n)*binomial(4*n, n+1) for n >= 1. (End)

Example

From Petros Hadjicostas, Jul 29 2020: (Start)
We give some examples to illustrate the comment by Sarah Selkirk about the total number of downs between the 1st and 2nd ups in a 2-Dyck path of length 4*(n+1). We denote by (+3) an up movement by a vector of (1,3) and by (-1) a down movement by a vector of (1,-1). We use powers to denote repetition of the same movement.
(i) For n = 0, we have the following 2-Dyck path of length 4 that contributes to a(0) = 3: (+3)(-1)^3 (no 2nd up here) with a total of 3 downs after the 1st up.
(ii) For n = 1, we have the following 2-Dyck paths of length 8 that contribute to a(1) = 6: (+3)(-1)(+3)(-1)^5, (+3)(-1)^2(+3)(-1)^4, and (+3)(-1)^3(+3)(-1)^3 with a contribution of 1 + 2 + 3 = 6 downs between the 1st and 2nd ups.
(iii) For n = 2, we have the following 2-Dyck paths of length 12 that contribute to a(2) = 28: (+3)(-1)(+3)(-1)^i(+3)(-1)^(8-i) for i = 0..5, (+3)(-1)^2(+3)(-1)^i(+3)^(7-i) for i = 0..4, and (+3)(-1)^3(+3)(-1)^i(+3)(-1)^(6-i) for i = 0..3 with a contribution of 1 x 6 + 2 x 5 + 3 x 4 = 28 downs between the 1st and 2nd ups. (End)

Mathematica

Table[3/(n+1) Binomial[4n, n], {n, 0, 30}] (* Harvey P. Dale, Nov 14 2013 *)

Prog

(PARI) a(n)={3*binomial(4*n, n)/(n+1)} \\ Andrew Howroyd, May 08 2020
(Magma) [3*Binomial(4*n, n)/(n+1) : n in [0..25]]; // Wesley Ivan Hurt, Jul 27 2020

Crossrefs

Cf. A002293, A007223, A007224, A007225, A007226, A007227, A007229, A124724, A263133, A006632.

Sequence in context: A370668 A220823 A024497 * A326074 A096155 A351069

Adjacent sequences: A007225 A007226 A007227 * A007229 A007230 A007231

Keyword

nonn,easy

Author

Simon Plouffe

Extensions

Edited by N. J. A. Sloane, Feb 07 2004 following a suggestion of Ralf Stephan

Reedited by N. J. A. Sloane, May 31 2008 following a suggestion of R. J. Mathar

Status

approved