0,3

Number of irreducible ordered pairs of compositions of n. A pair of compositions of n into the same number of (positive) parts, say n=a1+...+ak and n=b1+...+bk, is irreducible if for all j<k, a1+...+aj is not equal to b1+...+bj. E.g. a(3)=3 because the irreducible pairs are (1+2,2+1), (2+1,1+2), (3,3). - Herbert S. Wilf (wilf(AT)math.upenn.edu), May 22 2004

Hankel transform is 2^n. - Paul Barry (pbarry(AT)wit.ie), Nov 22 2007

Equals left border of triangle A152229 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]

Equals INVERTi transform of A000984: (1, 2, 6, 20, 70, 252,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]

Edward A. Bender, Gregory F. Lawler, Robin Pemantle and Herbert S. Wilf, Irreducible compositions and the first return to the origin of a random walk, preprint 2004

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Edward A. Bender, Gregory F. Lawler, Robin Pemantle and Herbert S. Wilf, Irreducible compositions and the first return to the origin of a random walk

G.f.: 1/(x+Sqrt[1-4x]). Recurrence: (n+3)y(n+3)-2(4n+9)y(n+2)+(15n+21)y(n+1)+2(2n+3)y(n) = 0

A Catalan transform of the Fibonacci numbers F(n+1) under the mapping G(x)-> G(xc(x)), c(x) the g.f. of A001008. The inverse mapping is H(x)->H(x(1-x)). a(n)=sum{k=0..n, (k/(2n-k))binomial(2n-k, n-k)F(k+1)} - Paul Barry (pbarry(AT)wit.ie), Dec 18 2004

G.f.: 1/(1-x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Aug 03 2009]

y[x_] := y[x] = (2(4n - 3)y[x - 1] - (15n - 24)y[x - 2] - (4n - 6)y[n - 3])/n y[0] = 1 y[1] = 1 y[2] = 3

Cf. A081698.

Cf. A000045.

A152229 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]

A000984 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]

Sequence in context: A082306 A124431 A071740 this_sequence A148939 A077587 A001893

Adjacent sequences: A081693 A081694 A081695 this_sequence A081697 A081698 A081699

easy,nonn

Emanuele Munarini (munarini(AT)mate.polimi.it), Apr 02 2003

More terms from Paul Barry (pbarry(AT)wit.ie), Dec 18 2004