cp's OEIS Frontend

A204456 Coefficient array of numerator polynomials of the o.g.f.s for the sequence of odd numbers not divisible by a given prime.

%I A204456 #17 Sep 01 2018 21:30:05
%S A204456 1,1,1,4,1,1,2,4,2,1,1,2,2,4,2,2,1,1,2,2,2,2,4,2,2,2,2,1,1,2,2,2,2,2,
%T A204456 4,2,2,2,2,2,1,1,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,1,1,2,2,2,2,2,2,2,2,4,
%U A204456 2,2,2,2,2,2,2,2,1,1,2,2,2,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,2,2,2,1
%N A204456 Coefficient array of numerator polynomials of the o.g.f.s for the sequence of odd numbers not divisible by a given prime.
%C A204456 The row length sequence of this array is p(m) = A000040(m) (the primes).
%C A204456 Row m, for m >= 1, lists the coefficients of the numerator polynomials N(p(m);x) = Sum_{k=0..p(m)-1} a(m,k)*x^k for the o.g.f. G(p(m);x) = x*N(p(m);x)/((1-x^(p(m)-1))*(1-x)) for the sequence a(p(m);n) of odd numbers not divisible by p(n). For m=1 one has a(2;n)=2*n-1, n >= 1, and for m > 1 one has a(p(m);n) = 2*n+1 + floor((n-(p(m)+1)/2)/(p(m)-1)), n >= 1, and a(p(m);0):=0. See A204454 for the m=5 sequence a(11;n), also for more details.
%C A204456 The rows of this array are symmetric. For m > 1 they are symmetric around the central 4.
%C A204456 The first (p(m)+1)/2 numbers of row number m, for m >= 2, are given by the first differences of the corresponding sequence {a(p(m);n)}, with a(p(m),0):=0. See a formula below. The proof is trivial for m=1, and clear for m >= 2 from a(p(m);n), for n=0,...,(p(m)+1)/2, which is {0,1,3,...,p-2,p+2}. - _Wolfdieter Lang_, Jan 26 2012
%F A204456 a(m,k) = [x^k]N(p(m);x), m>=1, k=0,...,p(m)-1, with the numerator polynomial N(p(m);x) for the o.g.f. G(p(m);x) of the sequence of odd numbers not divisible by the m-th prime p(m)=A000040(m). See the comment above.
%F A204456 Row m has the number pattern (exponents on a number indicate how many times this number appears consecutively):
%F A204456   m=1, p(1)=2: 1 1, and for m>=2:
%F A204456   m, p(m): 1 2^((p(m)-3)/2) 4 2^((p(m)-3)/2) 1.
%F A204456 a(m,k) = a(p(m);k+1) - a(p(m);k), m>=2, k=0,...,(p(m)-1)/2,
%F A204456 with the corresponding sequence {a(p(m);n)} of the odd numbers not divisible by p(m), with a(p(m);0):=0. For m=1: a(1,0) = a(2;1)-a(2;0). By symmetry around the center: a(m,(p(m)-1)/2+k) = a(m,(p(m)-1)/2-k), k=1,...,(p(m)-1)/2, m>=2. For m=1: a(1,1)=a(1,0). See a comment above. - _Wolfdieter Lang_, Jan 26 2012
%e A204456 The array starts
%e A204456 m,p\k  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
%e A204456 1,2:   1  1
%e A204456 2,3:   1  4  1
%e A204456 3,5:   1  2  4  2  1
%e A204456 4,7:   1  2  2  4  2  2  1
%e A204456 5,11:  1  2  2  2  2  4  2  2  2  2  1
%e A204456 6,13:  1  2  2  2  2  2  4  2  2  2  2  2  1
%e A204456 7,17:  1  2  2  2  2  2  2  2  4  2  2  2  2  2  2  2  1
%e A204456 ...
%e A204456 N(p(4);x) = N(7;x) = 1 + 2*x + 2*x^2 + 4*x^3 + 2*x^4 + 2*x^5 + x^6 = (1+x^2)*(1+2*x+x^2+2*x^3+x^4).
%e A204456 G(p(4);x) = G(7;x) = x*N(7;x)/((1-x^6)*(1-x)), the o.g.f. of
%e A204456 A162699. Compare this with the o.g.f. given there by _R. J. Mathar_, where the numerator is factorized also.
%e A204456 First difference rule: m=4: {a(7;n)} starts {0,1,3,5,9,...},
%e A204456 the first differences are {1,2,2,4,...}, giving the first (7+1)/2=4 entries of row number m=4 of the array. The other entries follow by symmetry. - _Wolfdieter Lang_, Jan 26 2012
%Y A204456 Cf. A000040, A005408 (p=2), A007310 (p=3), A045572 (p=5), A162699 (p=7), A204454 (p=11).
%K A204456 nonn,easy,tabf
%O A204456 1,4
%A A204456 _Wolfdieter Lang_, Jan 24 2012