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A026998 Triangular array T read by rows: T(n, k) = t(n, 2k), t given by A027960, 0 <= k <= n, n >= 0.

%I A026998 #21 Aug 22 2025 00:15:43
%S A026998 1,1,1,1,4,1,1,4,8,1,1,4,11,13,1,1,4,11,26,19,1,1,4,11,29,54,26,1,1,4,
%T A026998 11,29,73,101,34,1,1,4,11,29,76,171,174,43,1,1,4,11,29,76,196,370,281,
%U A026998 53,1,1,4,11,29,76,199,487,743,431,64,1
%N A026998 Triangular array T read by rows: T(n, k) = t(n, 2k), t given by A027960, 0 <= k <= n, n >= 0.
%C A026998 Right-edge columns are polynomials approximating Lucas(2n+1).
%H A026998 G. C. Greubel, <a href="/A026998/b026998.txt">Table of n, a(n) for n = 0..1325</a>
%F A026998 T(n, k) = Lucas(2*n+1) = A002878(n) for 2*k <= n, otherwise the (2*n-2*k)-th coefficient of the power series for (1+2*x)/( (1-x-x^2)*(1-x)^(2*k-n) ).
%e A026998   .................................... 1;
%e A026998   ................................. 1, 1;
%e A026998   ............................. 1,  4, 1;
%e A026998   ........................ 1,   4,  8, 1;
%e A026998   ................... 1,   4,  11, 13, 1;
%e A026998   .............. 1,   4,  11,  26, 19, 1;
%e A026998   .......... 1,  4,  11,  29,  54, 26, 1;
%e A026998   ...... 1,  4, 11,  29,  73, 101, 34, 1;
%e A026998   .. 1,  4, 11, 29,  76, 171, 174, 43, 1;
%e A026998   1, 4, 11, 29, 76, 196, 370, 281, 53, 1;
%t A026998 f[n_, k_]:= f[n, k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
%t A026998 A027960[n_, k_]:= LucasL[k+1] - f[n,k]*Boole[k>n];
%t A026998 A026998[n_, k_]:= A027960[n,2*k];
%t A026998 Table[A026998[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 09 2025 *)
%o A026998 (Magma)
%o A026998 function t(n, k) // t = A027960
%o A026998       if k le n then return Lucas(k+1);
%o A026998       elif k gt 2*n then return 0;
%o A026998       else return t(n-1, k-2) + t(n-1, k-1);
%o A026998       end if;
%o A026998 end function;
%o A026998 A026998:= func< n,k | t(n, 2*k) >;
%o A026998 [A026998(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 09 2025
%o A026998 (SageMath)
%o A026998 @CachedFunction
%o A026998 def t(n, k): # t = A027960
%o A026998     if (k>2*n): return 0
%o A026998     elif (k<n+1): return lucas_number2(k+1, 1, -1)
%o A026998     else: return t(n-1, k-2) + t(n-1, k-1)
%o A026998 def A026998(n,k): return t(n, 2*k)
%o A026998 print(flatten([[A026998(n, k) for k in (0..n)] for n in (0..12)])) # _G. C. Greubel_, Jul 09 2025
%Y A026998 This is a bisection of the "Lucas array" A027960, see A027011 for the other bisection.
%Y A026998 Row sums give A095121.
%Y A026998 Signed row sums give A090132.
%Y A026998 Diagonal sums give A027010.
%Y A026998 Right-edge columns include A034856, A027966, A027968, A027970, A027972.
%Y A026998 Cf. A000032.
%K A026998 nonn,tabl,changed
%O A026998 0,5
%A A026998 _Clark Kimberling_
%E A026998 Edited by _Ralf Stephan_, May 05 2005