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

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, 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, 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

Offset: 1

Author

Wolfdieter Lang, Jan 24 2012

Keywords

• nonn
• easy
• tabf

Comments

The row length sequence of this array is p(m) = A000040(m) (the primes).

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.

The rows of this array are symmetric. For m > 1 they are symmetric around the central 4.

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

Examples

			The array starts
m,p\k  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
1,2:   1  1
2,3:   1  4  1
3,5:   1  2  4  2  1
4,7:   1  2  2  4  2  2  1
5,11:  1  2  2  2  2  4  2  2  2  2  1
6,13:  1  2  2  2  2  2  4  2  2  2  2  2  1
7,17:  1  2  2  2  2  2  2  2  4  2  2  2  2  2  2  2  1
...
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).
G(p(4);x) = G(7;x) = x*N(7;x)/((1-x^6)*(1-x)), the o.g.f. of
A162699. Compare this with the o.g.f. given there by _R. J. Mathar_, where the numerator is factorized also.
First difference rule: m=4: {a(7;n)} starts {0,1,3,5,9,...},
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
		

Crossrefs

Cf. A000040, A005408 (p=2), A007310 (p=3), A045572 (p=5), A162699 (p=7), A204454 (p=11).

Formula

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.

Row m has the number pattern (exponents on a number indicate how many times this number appears consecutively):

m=1, p(1)=2: 1 1, and for m>=2:

m, p(m): 1 2^((p(m)-3)/2) 4 2^((p(m)-3)/2) 1.

a(m,k) = a(p(m);k+1) - a(p(m);k), m>=2, k=0,...,(p(m)-1)/2,

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