A092684 First column and main diagonal of triangle A092683, in which the convolution of each row with {1,1} produces a triangle that, when flattened, equals the flattened form of A092683.

1, 1, 2, 3, 6, 11, 21, 39, 74, 141, 272, 527, 1026, 2002, 3914, 7659, 14996, 29369, 57531, 112727, 220963, 433342, 850386, 1670011, 3282259, 6456475, 12711413, 25047465, 49396116, 97490480, 192552549, 380565123, 752619506, 1489234257

Offset: 0

Author

Paul D. Hanna, Mar 04 2004

Keywords

• nonn

Comments

The self-convolution forms A100938. - Paul D. Hanna, Nov 23 2004

The limit of the matrix power A011973^n, as n->inf, results in a single column vector equal to this sequence. - Paul D. Hanna, May 03 2006

Examples

			a(8) = Sum_{k=0..[8/2]} C(n-k,k)*a(k)
= C(8,0)*a(0) +C(7,1)*a(1) +C(6,2)*a(2) +C(5,3)*a(3) +C(4,4)*a(4)
= 1*1 + 7*1 + 15*2 + 10*3 + 1*6 = 74.
		

Crossrefs

Cf. A092683, A092685, A092686, A092689.

Cf. A011973 (Fibonacci polynomials), A100938 (self-convolution).

Programs

PARI

{T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,1, if(k==n,T(n,0), T(n-1,k)+T(n-1,k+1)))))}
a(n)=T(n,0)

PARI

a(n)=if(n==0,1,sum(k=0,n\2,binomial(n-k,k)*a(k))) \\ Paul D. Hanna, May 03 2006

PARI

{a(n)=local(A=1+x);for(i=0,n\2,A=subst(A,x,x^2/(1-x+x*O(x^n)))/(1-x));polcoeff(A,n)} \\ Paul D. Hanna, Jul 10 2006

Formula

Invariant under the transformation of Fibonacci triangle A011973(n,k)=C(n-k,k): a(n) = Sum_{k=0..[n/2]} C(n-k,k)*a(k). - Paul D. Hanna, May 03 2006

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*a(k). - Vladeta Jovovic, May 07 2006

G.f. satisfies: A(x) = A( x^2/(1-x) )/(1-x). - Paul D. Hanna, Jul 10 2006