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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.

%I A092684 #20 Apr 29 2025 17:18:51
%S A092684 1,1,2,3,6,11,21,39,74,141,272,527,1026,2002,3914,7659,14996,29369,
%T A092684 57531,112727,220963,433342,850386,1670011,3282259,6456475,12711413,
%U A092684 25047465,49396116,97490480,192552549,380565123,752619506,1489234257
%N 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.
%C A092684 The self-convolution forms A100938. - _Paul D. Hanna_, Nov 23 2004
%C A092684 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
%F A092684 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
%F A092684 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*a(k). - _Vladeta Jovovic_, May 07 2006
%F A092684 G.f. satisfies: A(x) = A( x^2/(1-x) )/(1-x). - _Paul D. Hanna_, Jul 10 2006
%e A092684 a(8) = Sum_{k=0..[8/2]} C(n-k,k)*a(k)
%e A092684 = 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)
%e A092684 = 1*1 + 7*1 + 15*2 + 10*3 + 1*6 = 74.
%o A092684 (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)))))}
%o A092684 a(n)=T(n,0)
%o A092684 (PARI) a(n)=if(n==0,1,sum(k=0,n\2,binomial(n-k,k)*a(k))) \\ _Paul D. Hanna_, May 03 2006
%o A092684 (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
%Y A092684 Cf. A092683, A092685, A092686, A092689.
%Y A092684 Cf. A011973 (Fibonacci polynomials), A100938 (self-convolution).
%K A092684 nonn
%O A092684 0,3
%A A092684 _Paul D. Hanna_, Mar 04 2004