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

0,3

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

Links

Table of n, a(n) for n=0..33.

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

Example

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.

Prog

(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

Crossrefs

Cf. A092683, A092685, A092686, A092689.

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

Sequence in context: A049856 A302017 A113409 * A366107 A123915 A132832

Adjacent sequences: A092681 A092682 A092683 * A092685 A092686 A092687

Keyword

nonn

Author

Paul D. Hanna, Mar 04 2004

Status

approved