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A117938 Triangle, columns generated from Lucas Polynomials.

%I A117938 #21 Oct 29 2021 11:37:02
%S A117938 1,1,1,1,2,3,1,3,6,4,1,4,11,14,7,1,5,18,36,34,11,1,6,27,76,119,82,18,
%T A117938 1,7,38,140,322,393,198,29,1,8,51,234,727,1364,1298,478,47,1,9,66,364,
%U A117938 1442,3775,5778,4287,1154,76,1,10,83,536,2599,8886,19602,24476,14159,2786,123
%N A117938 Triangle, columns generated from Lucas Polynomials.
%C A117938 Companion triangle using Fibonacci polynomial generators = A073133. Inverse binomial transforms of the columns defines rows of A117937 (with some adjustments of offset).
%C A117938 A309220 is another version of the same triangle (except it omits the last diagonal), and perhaps has a clearer definition. - _N. J. A. Sloane_, Aug 13 2019
%H A117938 G. C. Greubel, <a href="/A117938/b117938.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A117938 Columns are f(x), x = 1, 2, 3, ..., of the Lucas Polynomials: (1, defined different from A034807 and A114525); (x); (x^2 + 2); (x^3 + 3*x); (x^4 + 4*x^2 + 2); (x^5 + 5*x^3 + 5*x); (x^6 + 6*x^4 + 9*x^2 + 2); (x^7 + 7*x^5 + 14*x^3 + 7*x); ...
%e A117938 First few rows of the triangle are:
%e A117938   1;
%e A117938   1, 1;
%e A117938   1, 2,  3;
%e A117938   1, 3,  6,   4;
%e A117938   1, 4, 11,  14,   7;
%e A117938   1, 5, 18,  36,  34,  11;
%e A117938   1, 6, 27,  76, 119,  82,  18;
%e A117938   1, 7, 38, 140, 322, 393, 198, 29;
%e A117938   ...
%e A117938 For example, T(7,4) = 76 = f(4), x^3 + 3*x = 64 + 12 = 76.
%p A117938 Lucas := proc(n,x) # see A114525
%p A117938     option remember;
%p A117938     if  n=0 then
%p A117938         2;
%p A117938     elif n =1 then
%p A117938         x ;
%p A117938     else
%p A117938         x*procname(n-1,x)+procname(n-2,x) ;
%p A117938     end if;
%p A117938     expand(%) ;
%p A117938 end proc:
%p A117938 A117938 := proc(n::integer,k::integer)
%p A117938     if k = 1 then
%p A117938         1;
%p A117938     else
%p A117938         subs(x=n-k+1,Lucas(k-1,x)) ;
%p A117938     end if;
%p A117938 end proc:
%p A117938 seq(seq(A117938(n,k),k=1..n),n=1..12) ; # _R. J. Mathar_, Aug 16 2019
%t A117938 T[n_, k_]:= LucasL[k-1, n-k+1] - Boole[k==1];
%t A117938 Table[T[n, k], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Oct 28 2021 *)
%o A117938 (Sage)
%o A117938 def A117938(n,k): return 1 if (k==1) else round(2^(1-k)*( (n-k+1 + sqrt((n-k)*(n-k+2) + 5))^(k-1) + (n-k+1 - sqrt((n-k)*(n-k+2) + 5))^(k-1) ))
%o A117938 flatten([[A117938(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Oct 28 2021
%Y A117938 Cf. A104509, A114525, A117936, A117937, A118980, A118981, A309220.
%Y A117938 Cf. A000204 (diagonal), A059100 (column 3), A061989 (column 4).
%K A117938 nonn,tabl,easy
%O A117938 1,5
%A A117938 _Gary W. Adamson_, Apr 03 2006
%E A117938 Terms a(51) and a(52) corrected by _G. C. Greubel_, Oct 28 2021