login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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). 8
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 (list; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A083280 A060689 A053119 * A175267 A108045 A298972

Adjacent sequences:  A162512 A162513 A162514 * A162516 A162517 A162518

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Jul 05 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 12:17 EST 2021. Contains 349581 sequences. (Running on oeis4.)