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A162515 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = (U^n - L^n)/d, where U = (x + d)/2, L = (x - d)/2, d = sqrt(x^2 + 4).

%I A162515 #43 Sep 08 2022 08:45:46
%S A162515 0,1,1,0,1,0,1,1,0,2,0,1,0,3,0,1,1,0,4,0,3,0,1,0,5,0,6,0,1,1,0,6,0,10,
%T A162515 0,4,0,1,0,7,0,15,0,10,0,1,1,0,8,0,21,0,20,0,5,0,1,0,9,0,28,0,35,0,15,
%U A162515 0,1,1,0,10,0,36,0,56,0,35,0,6,0,1,0,11,0,45,0,84,0,70,0,21,0,1
%N A162515 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = (U^n - L^n)/d, where U = (x + d)/2, L = (x - d)/2, d = sqrt(x^2 + 4).
%C A162515 Row sums 0,1,1,2,3,5,... are the Fibonacci numbers, A000045.
%C A162515 Note that the coefficients are given in decreasing order. - _M. F. Hasler_, Dec 07 2011
%C A162515 Essentially a mirror image of A168561. - _Philippe Deléham_, Dec 08 2013
%H A162515 G. C. Greubel, <a href="/A162515/b162515.txt">Rows n = 0..101 of triangle</a>
%H A162515 T. Copeland, <a href="https://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>
%F A162515 P(n,x) = x*P(n-1, x) + P(n-2, x), where P(0,x)=0 and P(1,x)=1.
%F A162515 T(n,k) = T(n-1, k) + T(n-2, k-2) for n>=2. - _Philippe Deléham_, Dec 08 2013
%e A162515 Polynomial expansion:
%e A162515   0;
%e A162515   1;
%e A162515   x;
%e A162515   x^2 + 1;
%e A162515   x^3 + 2*x;
%e A162515   x^4 + 3*x^2 + 1;
%e A162515 First rows:
%e A162515   0;
%e A162515   1;
%e A162515   1, 0;
%e A162515   1, 0, 1;
%e A162515   1, 0, 2, 0;
%e A162515   1, 0, 3, 0, 1;
%e A162515   1, 0, 4, 0, 3, 0;
%e A162515 Row 6 matches P(6,x)=x^5 + 4*x^3 + 3*x.
%p A162515 0, seq(seq(`if`(`mod`(k,2)=0, binomial(n-k/2, k/2), 0), k = 0..n), n = 0..15); # _G. C. Greubel_, Jan 01 2020
%t A162515 Join[{0}, Table[If[EvenQ[k], Binomial[n-k/2, k/2], 0], {n,0,15}, {k,0,n} ]//Flatten] (* _G. C. Greubel_, Jan 01 2020 *)
%o A162515 (PARI) row(n,d=sqrt(1+x^2/4+O(x^n))) = Vec(if(n,Pol(((x/2+d)^n-(x/2-d)^n)/d)>>1)) \\ _M. F. Hasler_, Dec 07 2011, edited Jul 05 2021
%o A162515 (Magma)
%o A162515 function T(n,k)
%o A162515   if (k mod 2) eq 0 then return Round( Gamma(n-k/2+1)/(Gamma(k/2+1)*Gamma(n-k+1)));
%o A162515   else return 0;
%o A162515   end if; return T; end function;
%o A162515 [0] cat [T(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jan 01 2020
%o A162515 (Sage)
%o A162515 @CachedFunction
%o A162515 def T(n, k):
%o A162515     if (k%2==0): return binomial(n-k/2, k/2)
%o A162515     else: return 0
%o A162515 [0]+flatten([[T(n, k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jan 01 2020
%o A162515 (GAP)
%o A162515 T:= function(n,k)
%o A162515     if (k mod 2)=0 then return Binomial(n- k/2, k/2);
%o A162515     else return 0;
%o A162515     fi; end;
%o A162515 Concatenation([0], Flat(List([0..15], n-> List([0..n], k-> T(n,k) ))) ); # _G. C. Greubel_, Jan 01 2020
%Y A162515 Cf. A000045, A049310, A053119, A162514, A162516, A162517.
%K A162515 nonn,tabf
%O A162515 0,10
%A A162515 _Clark Kimberling_, Jul 05 2009