OFFSET
0,4
COMMENTS
A104975 is the sequence array for this sequence.
abs(a(n)) is the numbers of compositions of n into parts of the form 2^k-1, see example. [Joerg Arndt, Jan 06 2013]
LINKS
Robert Israel, Table of n, a(n) for n = 0..5509
Johann Cigler, Relation between a continued fraction and partitions, mathoverflow question 292231, Feb 05 2018.
Johann Cigler, A curious class of Hankel determinants, arXiv:1803.05164 [math.CO], 2018.
FORMULA
G.f.: x / (sum_{k>=0} x^(2^k)). (corrected by Joerg Arndt, Jan 06 2013)
G.f.: 1 / (1 + sum(n>=1, x^(2^n-1) ) ), replace the '+' by '-' to obtain the g.f. for compositions into parts 2^k-1. [Joerg Arndt, Jan 06 2013]
G.f.: 1 - x / (1 + x / (1 + x / (1 - x / (1 + x / (1 - x / ...))))) = 1 + b(1) * x / (1 + b(2) * x / (1 + b(3) * x / ...)) where b(n) = (-1)^ A090678(n+1) [Conjecture]. - Michael Somos, Jan 03 2013
Convolution inverse is A209229 with 0 preprended. - Michael Somos, Jan 03 2013
a(n) = sum(m=0..n, T(n,m)*(-1)^(m)), where T(n,m)=(1+(-1)^(n-m))/2 *sum(k=1..(n-m)/2, binomial(m,k)*T((n-m)/2,k)), T(n,n)=1. - Vladimir Kruchinin, Mar 18 2015
EXAMPLE
G.f. = 1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 6*x^6 - 10*x^7 + 15*x^8 - 22*x^9 + 34*x^10 + ...
From Joerg Arndt, Jan 06 2013: (Start)
There are abs(a(8)) = 15 compositions of 8 into parts 2^k-1:
[ 1] [ 1 1 1 1 1 1 1 1 ]
[ 2] [ 1 1 1 1 1 3 ]
[ 3] [ 1 1 1 1 3 1 ]
[ 4] [ 1 1 1 3 1 1 ]
[ 5] [ 1 1 3 1 1 1 ]
[ 6] [ 1 1 3 3 ]
[ 7] [ 1 3 1 1 1 1 ]
[ 8] [ 1 3 1 3 ]
[ 9] [ 1 3 3 1 ]
[10] [ 1 7 ]
[11] [ 3 1 1 1 1 1 ]
[12] [ 3 1 1 3 ]
[13] [ 3 1 3 1 ]
[14] [ 3 3 1 1 ]
[15] [ 7 1 ]
(End)
MAPLE
N:= 100: # to get a(0)..a(N)
S:= series(x/add(x^(2^k), k=0..ilog2(N+1)), x, N+2):
[seq](coeff(S, x, j), j = 0 .. N); # Robert Israel, Feb 07 2018
MATHEMATICA
T[n_, n_] = 1; T[n_, m_] := T[n, m] =(1 + (-1)^(n-m))/2 Sum[Binomial[m, k]* T[(n-m)/2, k], {k, 1, (n-m)/2}];
a[n_] := Sum[T[n, m] (-1)^m, {m, 0, n}];
Array[a, 50, 0] (* Jean-François Alcover, Jul 21 2018, after Vladimir Kruchinin *)
PROG
(PARI)
N = 66; q = 'q + O('q^N); L = 2+ceil(log(N)/log(2));
gf = q / sum(n=0, L, q^(2^n) );
/* gf = 1 / (1 + sum(n=1, L, q^(2^n-1) ) ); */ /* same */
v = Vec(gf)
/* Joerg Arndt, Jan 06 2013 */
(Maxima)
T(n, m):=if n=m then 1 else (1+(-1)^(n-m))/2*sum(binomial(m, k)*T((n-m)/2, k), k, 1, (n-m)/2); makelist(sum(T(n, m)*(-1)^(m), m, 0, n), n, 0, 20); /* Vladimir Kruchinin, Mar 18 2015 */
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Mar 30 2005
EXTENSIONS
Added more terms, Joerg Arndt, Jan 06 2013
STATUS
approved