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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 (list; table; graph; refs; listen; history; text; internal format)
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 * A123915 A132832 A316796

Adjacent sequences:  A092681 A092682 A092683 * A092685 A092686 A092687

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 04 2004

STATUS

approved

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Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)