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A115322 Triangle of coefficients of Pell polynomials.

%I A115322 #31 Aug 07 2025 03:08:20
%S A115322 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,
%T A115322 80,0,192,0,128,1,0,40,0,240,0,448,0,256,0,10,0,160,0,672,0,1024,0,
%U A115322 512,1,0,60,0,560,0,1792,0,2304,0,1024,0,12,0,280,0,1792,0,4608,0,5120,0,2048
%N A115322 Triangle of coefficients of Pell polynomials.
%C A115322 Aside from signs, same as A053117.
%C A115322 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
%H A115322 Jonathan L. Gross and Thomas W. Tucker, <a href="https://doi.org/10.1007/s00454-010-9257-0">A Celtic Framework for Knots and Links</a>, Discrete & Computational Geometry 46 (2011), 86-99.
%H A115322 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellPolynomial.html">Pell Polynomial</a>
%F A115322 G.f. for n-th row is Fibonacci(n, 2*x).
%F A115322 From _Franck Maminirina Ramaharo_, Aug 06 2025: (Start)
%F A115322 Row n = coefficients in the expansion of ((x + srt(x^2 + 1))^n - (x - sqrt(x^2 + 1))^n)/(2*sqrt(x^2 + 1)).
%F A115322 T(n,k) = 0 if n == k (mod 2) else T(n,k) = (2^k)*binomial((n + k - 1)/2, k).
%F A115322 G.f.: y/(1 - 2 x*y - y^2).
%F A115322 (End)
%e A115322 1, 2*x, 1 + 4*x^2, 4*x + 8*x^3, 1 + 12*x^2 + 16*x^4, ...
%e A115322 Triangle begins:
%e A115322   [n\k] 0   1   2    3    4    5     6    7    8     9    10
%e A115322   ----------------------------------------------------------
%e A115322   [1 ]  1;
%e A115322   [2 ]  0,  2;
%e A115322   [3 ]  1,  0,  4;
%e A115322   [4 ]  0,  4,  0,   8;
%e A115322   [5 ]  1,  0, 12,   0,  16;
%e A115322   [6 ]  0,  6,  0,  32,   0,  32;
%e A115322   [7 ]  1,  0, 24,   0,  80,   0,   64;
%e A115322   [8 ]  0,  8,  0,  80,   0, 192,    0,  128;
%e A115322   [9 ]  1,  0, 40,   0, 240,   0,  448,    0,  256;
%e A115322   [10]  0, 10,  0, 160,   0, 672,    0, 1024,    0, 512;
%e A115322   [11]  1,  0, 60,   0, 560,   0, 1792,    0, 2304,   0, 1024;
%e A115322 ... - _Franck Maminirina Ramaharo_, Aug 06 2025
%p A115322 A115322 := (n, k) -> ifelse(irem(n - k, 2) = 0, 0, binomial((n + k - 1)/2, k)*2^k);
%p A115322 seq(print(seq(A115322(n, k), k = 0..n-1)), n = 1..10);  # _Peter Luschny_, Aug 06 2025
%t A115322 Flatten[Table[CoefficientList[Fibonacci[n, 2 x], x], {n, 0, 20}]] (* _Emanuele Munarini_, Dec 01 2017 *)
%o A115322 (Maxima)
%o A115322 T(n, k) := if mod(n - k, 2) = 0 then 0 else  binomial((n + k - 1)/2, k)*2^k$
%o A115322 create_list(T(n, k), n, 1, 10, k, 0, n - 1); /* _Franck Maminirina Ramaharo_, Aug 06 2025 */
%Y A115322 Cf. A053117, A049310, A000129 (row sums).
%K A115322 nonn,tabl
%O A115322 1,3
%A A115322 _Eric W. Weisstein_, Jan 20 2006