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A007223 Number of distinct perforation patterns for deriving (v,b) = (n+2,n) punctured convolutional codes from (2,1).

%I A007223 M1843 #49 Apr 30 2023 07:10:14
%S A007223 1,2,8,24,85,286,1008,3536,12618,45220,163504,594320,2173197,7983990,
%T A007223 29465440,109174560,405995326,1514797020,5669021488,21275014800,
%U A007223 80047272578,301892460012,1141069157408,4321730134624,16399422757300
%N A007223 Number of distinct perforation patterns for deriving (v,b) = (n+2,n) punctured convolutional codes from (2,1).
%C A007223 Number of ways to distribute n pairs parentheses into 2 groups where each group of parentheses represents a Catalan ordering (A000108), and each group must contain at least one pair of parentheses. If one of the groups may have no parentheses, we arrive at A007595. Analog of A274934 with Catalan numbers replacing connected graph counts. - _R. J. Mathar_, Jul 19 2016
%C A007223 From _Petros Hadjicostas_, Jul 27 2020: (Start)
%C A007223 "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).)
%C A007223 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).
%C A007223 Even though the 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 program below. (End)
%D A007223 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.
%D A007223 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007223 Guy Bégin and David Haccoun, <a href="https://doi.org/10.1109/26.44210">High rate punctured convolutions codes: Structure properties and construction techniques</a>, IEEE Transactions on Communications 37(12) (1989), 1381-1385.
%H A007223 David Haccoun and Guy Bégin, <a href="https://doi.org/10.1109/26.46505">High rate punctured convolutional codes for Viterbi and sequential coding</a>, IEEE Transactions on Communications, 37(11) (1989), 1113-1125; see Section II.
%F A007223 Conjecture: Expansion of [A(x)^2 + A(x^2)]/2, where A(x) = A000108(x) - 1. - _R. J. Mathar_, Jul 19 2016
%F A007223 From _Petros Hadjicostas_, Jul 27 2020: (Start)
%F A007223 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).
%F A007223 According to Pab Ter's Maple code, this is the coefficient of z^v in the polynomial (1/b)*Sum_{k|b} phi(k)*(1 + z^k)^(v0*b/k).
%F A007223 Here (v,b) = (n+2,n) and (v0,1) = (2,1), so
%F A007223 a(n) = (1/n)*Sum_{k|gcd(n+2,n)} phi(k)*binomial(2*n/k, (n+2)/k).
%F A007223 This simplifies to
%F A007223 a(n) = (1/n)*(binomial(2*n, n+2) + [(n mod 2) == 0]*binomial(n, (n/2) + 1)).
%F A007223 It follows from my comments in A275206 that _R. J. Mathar_'s conjecture is correct and that
%F A007223 a(n) = (-2*c(n) + c(n+1) + [(n mod 2) == 0]*c(n/2))/2 for n >= 1, where c = A000108. (End)
%F A007223 D-finite with recurrence -(11*n-30)*(n+2)*(n+1) *a(n) +10*(n+1) *(7*n^2-22*n+6) *a(n-1) -60*(n-2)*(n^2-5*n+1) *a(n-2) -40*(n-2) *(7*n^2-22*n+6) *a(n-3) +16*(2*n-7) *(n-3) *(13*n-22) *a(n-4)=0. - _R. J. Mathar_, Mar 21 2021
%p A007223 with(numtheory):P:=proc(b,v0) local k: RETURN(add(phi(k)*(1+z^k)^(v0*(b/k)),k=divisors(b))/b): end; seq(coeff(P(b,2),z,b+2),b=2..40); # Pab Ter
%t A007223 A[x_] = (1 - Sqrt[1 - 4x])/(2x) - 1;
%t A007223 CoefficientList[(A[x]^2 + A[x^2])/(2 x^2) + O[x]^25, x] (* _Jean-François Alcover_, Apr 30 2023, after _R. J. Mathar_'s proven conjecture *)
%Y A007223 Cf. A000108, A007229, A007595, A274934, A275206.
%K A007223 nonn
%O A007223 2,2
%A A007223 _Simon Plouffe_
%E A007223 More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005
%E A007223 a(2) = 1 prepended by _R. J. Mathar_, Jul 19 2016