

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.


8



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
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OFFSET

0,3


COMMENTS

The selfconvolution 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(nk,k): a(n) = Sum_{k=0..[n/2]} C(nk,k)*a(k).  Paul D. Hanna, May 03 2006
a(n) = Sum_{k=0..floor(n/2)} binomial(nk,k)*a(k).  Vladeta Jovovic, May 07 2006
G.f. satisfies: A(x) = A( x^2/(1x) )/(1x).  Paul D. Hanna, Jul 10 2006


EXAMPLE

a(8) = Sum_{k=0..[8/2]} C(nk,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(n1, k)+T(n1, k+1)))))} a(n)=T(n, 0)
(PARI) a(n)=if(n==0, 1, sum(k=0, n\2, binomial(nk, 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/(1x+x*O(x^n)))/(1x)); polcoeff(A, n)}  Paul D. Hanna, Jul 10 2006


CROSSREFS

Cf. A092683, A092685, A092686, A092689.
Cf. A011973 (Fibonacci polynomials), A100938 (selfconvolution).
Sequence in context: A049856 A302017 A113409 * A123915 A132832 A316796
Adjacent sequences: A092681 A092682 A092683 * A092685 A092686 A092687


KEYWORD

nonn,tabl


AUTHOR

Paul D. Hanna, Mar 04 2004


STATUS

approved



