A227635 - OEIS

%I #20 Oct 28 2014 16:33:34

%S 1,3,5,8,12,18,28,42,65,103,160,252,404,644,1026,1648,2654,4269,6873,

%T 11086,17898,28902,46681,75427,121920,197116,318731,515425,833593,

%U 1348316,2181023,3528149,5707571,9233632,14938484,24168539,39102325,63264687,102358843,165612728

%N G.f.: Sum_{n>=1} x^n * (1+x)^n / (1-x^n).

%C a(n) = column sums in an array with rows consisting of n' compositions into X and (X+1) parts; n'>0, X=1...inf.

%C 1 1 2 3 5 8 13 21 34 55 89...

%C 1 0 1 1 1 2  2  3  4  5  7

%C 1 0 0 1 1 0  1  2  1  1  3

%C 1 0 0 0 1 1  0  0  1  2  1

%C 1 0 0 0 0 1  1  0  0  0  1

%C 1 0 0 0 0 0  1  1  0  0  0

%C 1 0 0 0 0 0  0  1  1  0  0

%C 1 0 0 0 0 0  0  0  1  1  0

%C 1 0 0 0 0 0  0  0  0  1  1

%C 1 0 0 0 0 0  0  0  0  0  1...

%C +=1 3 5 8 12 18 28 42 65 103...

%C - _Bob Selcoe_, Feb 07 2014

%H Vaclav Kotesovec, <a href="/A227635/b227635.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum_{n>=1} x^n * Sum_{d|n} (1+x)^d.

%F a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n+1). - _Vaclav Kotesovec_, Oct 28 2014

%e G.f.: A(x) = x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 18*x^6 + 28*x^7 + 42*x^8 +...

%e where

%e A(x) = x*(1+x)/(1-x) + x^2*(1+x)^2/(1-x^2) + x^3*(1+x)^3/(1-x^3) + x^4*(1+x)^4/(1-x^4) + x^5*(1+x)^5/(1-x^5) + x^6*(1+x)^6/(1-x^6) +...

%o (PARI) {a(n)=polcoeff(sum(m=1,n,x^m*(1+x)^m/(1-x^m +x*O(x^n)) ),n)}

%o for(n=1,40,print1(a(n),", "))

%o (PARI) {a(n)=polcoeff(sum(m=1,n,x^m*sumdiv(m,d,(1+x +x*O(x^n))^d) ),n)}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A097939.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 18 2013