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A007223
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Number of distinct perforation patterns for deriving (v,b) = (n+2,n) punctured convolutional codes from (2,1).
(Formerly M1843)
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7
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1, 2, 8, 24, 85, 286, 1008, 3536, 12618, 45220, 163504, 594320, 2173197, 7983990, 29465440, 109174560, 405995326, 1514797020, 5669021488, 21275014800, 80047272578, 301892460012, 1141069157408, 4321730134624, 16399422757300
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OFFSET
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2,2
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COMMENTS
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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
"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 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)
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REFERENCES
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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).
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LINKS
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FORMULA
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Conjecture: Expansion of [A(x)^2 + A(x^2)]/2, where A(x) = A000108(x) - 1. - R. J. Mathar, Jul 19 2016
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, 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+2,n) and (v0,1) = (2,1), so
a(n) = (1/n)*Sum_{k|gcd(n+2,n)} phi(k)*binomial(2*n/k, (n+2)/k).
This simplifies to
a(n) = (1/n)*(binomial(2*n, n+2) + [(n mod 2) == 0]*binomial(n, (n/2) + 1)).
a(n) = (-2*c(n) + c(n+1) + [(n mod 2) == 0]*c(n/2))/2 for n >= 1, where c = A000108. (End)
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
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MAPLE
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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
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MATHEMATICA
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A[x_] = (1 - Sqrt[1 - 4x])/(2x) - 1;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005
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STATUS
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approved
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