1,3

This sequence convolved with A000217 (without initial term 0) yields A164680.

See A164680 for similar convolutions.

Contribution from Alford Arnold (Alford1940(AT)aol.com), Sep 24 2009: (Start)

A165188 convolved with A000217 yields sequence A164680. This is to be expected

since A000217 can be associated with partition 1+1+1, A164680 with partition

1+1+1+2+2+2+3 and A165188 with partition 2+2+2+3 by observing their unreduced

generating functions and verified by generating the sequences

by converting the partitions into finite sequences and using Euler's

Transform.

Thus partition 1+1+1 yields the finite sequence (3);

partition 2+2+2+3 yields the finite sequence (0,3,1);

and, when combined, partition 1+1+1+2+2+2+3 yields (3,3,1).

(End)

a(n) = -a(n-1)+2*a(n-2)+3*a(n-3)-3*a(n-5)-2*a(n-6)+a(n-7)+a(n-8)+1 for n > 8; a(1)=1, a(2)=0, a(3)=3, a(4)=1, a(5)=6, a(6)=3, a(7)=11, a(8)=6. [From Klaus Brockhaus, Sep 15 2009]

G.f.: (x/((1-x)^4*(1+x)^3*(1+x+x^2)). [From Klaus Brockhaus, Sep 15 2009]

a(n)= (2*n^3+21*n^2+63*n+49)/288-(-1)^n*(9+7*n+n^2)/32+A057078(n)/9. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 17 2009]

A014125 begins 1,3,6,11,18,27,... , thus this sequence begins 1,0,3,1,6,3,11,6,18,11,27,18,... .

(PARI) /* first computes u = A014125 as second bisection of A001400, then interleaves */ {m=28; u=vector(m, n, polcoeff(1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))+O(x^(2*n)), 2*n-1)); vector(2*m, k, if(k%2==1 , u[(k+1)/2], if(k==2, 0, u[k/2-1])))} [From Klaus Brockhaus, Sep 15 2009]

Cf. A014125, A000217, A164680, A001400.

Sequence in context: A158822 A121443 A008795 this_sequence A132180 A126191 A070883

Adjacent sequences: A165185 A165186 A165187 this_sequence A165189 A165190 A165191

nonn

Alford Arnold (Alford1940(AT)aol.com), Sep 13 2009

Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 15 2009