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A007227 Number of distinct perforation patterns for deriving (v,b) = (n+2,n) punctured convolutional codes from (3,1).

%I A007227 M4624 #28 Jul 28 2020 02:47:22
%S A007227 9,42,236,1287,7314,41990,245256,1448655,8649823,52106040,316360752,
%T A007227 1933910820,11893566078,73537906926,456864894288,2850557192175,
%U A007227 17854854154215,112230508880490,707714010205020,4475876883386895
%N A007227 Number of distinct perforation patterns for deriving (v,b) = (n+2,n) punctured convolutional codes from (3,1).
%C A007227 From _Petros Hadjicostas_, Jul 27 2020: (Start)
%C A007227 "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 A007227 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 A007227 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 A007227 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 A007227 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007227 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 A007227 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 A007227 From _Petros Hadjicostas_, Jul 27 2020: (Start)
%F A007227 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 A007227 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 A007227 Here (v,b) = (n+2,n) and (v0,1) = (3,1), so
%F A007227 a(n) = (1/n)*Sum_{k|gcd(n+2,n)} phi(k)*binomial(3*n/k, (n+2)/k).
%F A007227 This simplifies to
%F A007227 a(n) = (1/n)*(binomial(3*n, n+2) + [(n mod 2) = 0]*binomial(3*n/2, (n/2) + 1)), where [ ] is the Iverson bracket. (End)
%p A007227 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,3),z,b+2),b=2..40); (Pab Ter)
%Y A007227 Cf. A007223, A007224, A007225, A007229.
%K A007227 nonn
%O A007227 2,1
%A A007227 _Simon Plouffe_
%E A007227 More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005