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

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, 11, 29, 73, 101, 34, 1, 1, 4, 11, 29, 76, 171, 174, 43, 1, 1, 4, 11, 29, 76, 196, 370, 281, 53, 1, 1, 4, 11, 29, 76, 199, 487, 743, 431, 64, 1

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl
• changed

Comments

Right-edge columns are polynomials approximating Lucas(2n+1).

Examples

			  .................................... 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, 11,  29,  73, 101, 34, 1;
  .. 1,  4, 11, 29,  76, 171, 174, 43, 1;
  1, 4, 11, 29, 76, 196, 370, 281, 53, 1;
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1325

Crossrefs

This is a bisection of the "Lucas array" A027960, see A027011 for the other bisection.

Row sums give A095121.

Signed row sums give A090132.

Diagonal sums give A027010.

Right-edge columns include A034856, A027966, A027968, A027970, A027972.

Cf. A000032.

Programs

Magma

function t(n, k) // t = A027960
      if k le n then return Lucas(k+1);
      elif k gt 2*n then return 0;
      else return t(n-1, k-2) + t(n-1, k-1);
      end if;
end function;
A026998:= func< n,k | t(n, 2*k) >;
[A026998(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2025

Mathematica

f[n_, k_]:= f[n, k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
A027960[n_, k_]:= LucasL[k+1] - f[n,k]*Boole[k>n];
A026998[n_, k_]:= A027960[n,2*k];
Table[A026998[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 09 2025 *)

SageMath

@CachedFunction
def t(n, k): # t = A027960
    if (k>2*n): return 0
    elif (kA026998(n,k): return t(n, 2*k)
print(flatten([[A026998(n, k) for k in (0..n)] for n in (0..12)])) # G. C. Greubel, Jul 09 2025

Formula

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) ).

Extensions

Edited by Ralf Stephan, May 05 2005