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).

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, 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, 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

Offset: 0

Author

Clark Kimberling, Jul 05 2009

Keywords

• nonn
• tabf

Comments

Row sums 0,1,1,2,3,5,... are the Fibonacci numbers, A000045.

Note that the coefficients are given in decreasing order. - M. F. Hasler, Dec 07 2011

Essentially a mirror image of A168561. - Philippe Deléham, Dec 08 2013

Examples

			Polynomial expansion:
  0;
  1;
  x;
  x^2 + 1;
  x^3 + 2*x;
  x^4 + 3*x^2 + 1;
First rows:
  0;
  1;
  1, 0;
  1, 0, 1;
  1, 0, 2, 0;
  1, 0, 3, 0, 1;
  1, 0, 4, 0, 3, 0;
Row 6 matches P(6,x)=x^5 + 4*x^3 + 3*x.
		

Links

• G. C. Greubel, Rows n = 0..101 of triangle
• T. Copeland, Addendum to Elliptic Lie Triad

Crossrefs

Cf. A000045, A049310, A053119, A162514, A162516, A162517.

Programs

GAP

T:= function(n,k)
    if (k mod 2)=0 then return Binomial(n- k/2, k/2);
    else return 0;
    fi; end;
Concatenation([0], Flat(List([0..15], n-> List([0..n], k-> T(n,k) ))) ); # G. C. Greubel, Jan 01 2020

Magma

function T(n,k)
  if (k mod 2) eq 0 then return Round( Gamma(n-k/2+1)/(Gamma(k/2+1)*Gamma(n-k+1)));
  else return 0;
  end if; return T; end function;
[0] cat [T(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 01 2020

Maple

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

Mathematica

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 *)

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

Sage

@CachedFunction
def T(n, k):
    if (k%2==0): return binomial(n-k/2, k/2)
    else: return 0
[0]+flatten([[T(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jan 01 2020

Formula

P(n,x) = x*P(n-1, x) + P(n-2, x), where P(0,x)=0 and P(1,x)=1.

T(n,k) = T(n-1, k) + T(n-2, k-2) for n>=2. - Philippe Deléham, Dec 08 2013