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