OFFSET
0,10
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
LINKS
G. C. Greubel, Rows n = 0..101 of triangle
T. Copeland, Addendum to Elliptic Lie Triad
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
EXAMPLE
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.
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 *)
PROG
(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
(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
(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
(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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Jul 05 2009
STATUS
approved