A115322 Triangle of coefficients of Pell polynomials.

1, 0, 2, 1, 0, 4, 0, 4, 0, 8, 1, 0, 12, 0, 16, 0, 6, 0, 32, 0, 32, 1, 0, 24, 0, 80, 0, 64, 0, 8, 0, 80, 0, 192, 0, 128, 1, 0, 40, 0, 240, 0, 448, 0, 256, 0, 10, 0, 160, 0, 672, 0, 1024, 0, 512, 1, 0, 60, 0, 560, 0, 1792, 0, 2304, 0, 1024, 0, 12, 0, 280, 0, 1792, 0, 4608, 0, 5120, 0, 2048

Offset: 1

Author

Eric W. Weisstein, Jan 20 2006

Keywords

• nonn
• tabl

Comments

Aside from signs, same as A053117.

Row n gives the coefficients in the expansion of the Alexander-Conway polynomial for the barrier-free Celtic link CK_4^(2n). - Franck Maminirina Ramaharo, Aug 06 2025

Examples

			1, 2*x, 1 + 4*x^2, 4*x + 8*x^3, 1 + 12*x^2 + 16*x^4, ...
Triangle begins:
  [n\k] 0   1   2    3    4    5     6    7    8     9    10
  ----------------------------------------------------------
  [1 ]  1;
  [2 ]  0,  2;
  [3 ]  1,  0,  4;
  [4 ]  0,  4,  0,   8;
  [5 ]  1,  0, 12,   0,  16;
  [6 ]  0,  6,  0,  32,   0,  32;
  [7 ]  1,  0, 24,   0,  80,   0,   64;
  [8 ]  0,  8,  0,  80,   0, 192,    0,  128;
  [9 ]  1,  0, 40,   0, 240,   0,  448,    0,  256;
  [10]  0, 10,  0, 160,   0, 672,    0, 1024,    0, 512;
  [11]  1,  0, 60,   0, 560,   0, 1792,    0, 2304,   0, 1024;
... - _Franck Maminirina Ramaharo_, Aug 06 2025
		

Links

• Jonathan L. Gross and Thomas W. Tucker, A Celtic Framework for Knots and Links, Discrete & Computational Geometry 46 (2011), 86-99.
• Eric Weisstein's World of Mathematics, Pell Polynomial

Crossrefs

Cf. A053117, A049310, A000129 (row sums).

Programs

Maple

A115322 := (n, k) -> ifelse(irem(n - k, 2) = 0, 0, binomial((n + k - 1)/2, k)*2^k);
seq(print(seq(A115322(n, k), k = 0..n-1)), n = 1..10);  # Peter Luschny, Aug 06 2025

Mathematica

Flatten[Table[CoefficientList[Fibonacci[n, 2 x], x], {n, 0, 20}]] (* Emanuele Munarini, Dec 01 2017 *)

Maxima

T(n, k) := if mod(n - k, 2) = 0 then 0 else  binomial((n + k - 1)/2, k)*2^k$
create_list(T(n, k), n, 1, 10, k, 0, n - 1); /* Franck Maminirina Ramaharo, Aug 06 2025 */

Formula

G.f. for n-th row is Fibonacci(n, 2*x).

From Franck Maminirina Ramaharo, Aug 06 2025: (Start)

Row n = coefficients in the expansion of ((x + srt(x^2 + 1))^n - (x - sqrt(x^2 + 1))^n)/(2*sqrt(x^2 + 1)).

T(n,k) = 0 if n == k (mod 2) else T(n,k) = (2^k)*binomial((n + k - 1)/2, k).

G.f.: y/(1 - 2 x*y - y^2).

(End)