A227635 G.f.: Sum_{n>=1} x^n * (1+x)^n / (1-x^n).

1, 3, 5, 8, 12, 18, 28, 42, 65, 103, 160, 252, 404, 644, 1026, 1648, 2654, 4269, 6873, 11086, 17898, 28902, 46681, 75427, 121920, 197116, 318731, 515425, 833593, 1348316, 2181023, 3528149, 5707571, 9233632, 14938484, 24168539, 39102325, 63264687, 102358843, 165612728

Offset: 1

Author

Paul D. Hanna, Jul 18 2013

Keywords

• nonn

Comments

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

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

1 0 1 1 1 2 2 3 4 5 7

1 0 0 1 1 0 1 2 1 1 3

1 0 0 0 1 1 0 0 1 2 1

1 0 0 0 0 1 1 0 0 0 1

1 0 0 0 0 0 1 1 0 0 0

1 0 0 0 0 0 0 1 1 0 0

1 0 0 0 0 0 0 0 1 1 0

1 0 0 0 0 0 0 0 0 1 1

1 0 0 0 0 0 0 0 0 0 1...

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

- Bob Selcoe, Feb 07 2014

Examples

			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 +...
where
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) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A097939.

Programs

PARI

{a(n)=polcoeff(sum(m=1,n,x^m*(1+x)^m/(1-x^m +x*O(x^n)) ),n)}
for(n=1,40,print1(a(n),", "))

PARI

{a(n)=polcoeff(sum(m=1,n,x^m*sumdiv(m,d,(1+x +x*O(x^n))^d) ),n)}
for(n=1,40,print1(a(n),", "))

Formula

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

a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n+1). - Vaclav Kotesovec, Oct 28 2014