Search: id:A094440
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%I A094440
%S A094440 1,1,2,2,3,3,3,8,6,4,5,15,20,10,5,8,30,45,40,15,6,13,56,105,105,70,21,
               7,
%T A094440 21,104,224,280,210,112,28,8,34,189,468,672,630,378,168,36,9,55,340,945,
%U A094440 1560,1680,1260,630,240,45,10,89,605,1870,3465,4290,3696,2310,990,330
%N A094440 Triangular array T(n,k)=F(n+1-k)C(n,k-1), k=1,2,3,...,n; n>=1.
%C A094440 Row sums yield the even-subscripted Fibonacci numbers (A001906).
%F A094440 Comments from Peter Bala (pbala(AT)toucansurf.com), Aug 17 2007: (Start)
%F A094440 The row polynomials F(n,x) = sum {k = 0..n} C(n,k)*Fibonacci(n-k)*x^k 
               satisfy F(n,x)* L(n,x) = F(2n,x), where L(n,x) = sum {k = 0..n} C(n,
               k)*Lucas(n-k)*x^k.
%F A094440 Other identities and formulas include:
%F A094440 F(n+1,x)^2 - F(n,x)F(n+2,x) = (x^2 + x - 1)^n;
%F A094440 Sum {k = 0..n}C(n,k)*F(n-k,x)*L(k,x) = 2^n F(n,x);
%F A094440 F(n,2x) = sum {k = 0..n} C(n,k)*F(n-k,x)*x^k;
%F A094440 F(n,3x) = sum {k = 0..n} C(n,k)*F(n-k,2x)*x^k etc;
%F A094440 Sequence {F(n,r)} n>=1 gives the r th binomial transform of the Fibonacci 
               numbers: r=1 gives A001906, r=2 gives A030191, r=3 gives A099453, 
               r=4 gives A081574, r=5 gives A081574.
%F A094440 F(n,1/phi) = (-1)^(n-1) F(n,-phi) = sqrt(5)^(n-1) for n >= 1, where phi 
               = (1+sqrt(5))/2.
%F A094440 The polynomials F(n,-x) satisfy a Riemann hypothesis: the zeros of F(n,
               -x) lie on the vertical line Re x = 1/2 in the complex plane.
%F A094440 G.f.: t/(1 - (2x + 1)*t + (x^2 + x - 1)*t^2) = t + (1 + 2x)*t^2 + (2 
               + 3x + 3x^2)*t^3 + (3 + 8x + 6x^2 + 4x^3)*t^4 + ... . (End)
%e A094440 Triangle starts:
%e A094440 1;
%e A094440 1, 2;
%e A094440 2, 3, 3;
%e A094440 3, 8, 6, 4;
%e A094440 T(4,3)=F(2)*C(4,2)=1*6=6
%p A094440 with(combinat): T:=(n,k)->binomial(n,k-1)*fibonacci(n+1-k): for n from 
               1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular 
               form (Deutsch)
%Y A094440 Cf. A094435, A000045.
%Y A094440 Cf. A001906.
%Y A094440 Cf. A132148.
%Y A094440 Sequence in context: A025496 A099959 A099964 this_sequence A093736 A076938 
               A014589
%Y A094440 Adjacent sequences: A094437 A094438 A094439 this_sequence A094441 A094442 
               A094443
%K A094440 nonn,tabl
%O A094440 1,3
%A A094440 Clark Kimberling (ck6(AT)evansville.edu), May 03 2004
%E A094440 Corrected error in expansion of generating function. - Peter Bala (pbala(AT)toucansurf.com), 
               Sep 24 2008

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