A112743 An aerated Delannoy triangle.

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 5, 0, 1, 0, 5, 0, 7, 0, 1, 1, 0, 13, 0, 9, 0, 1, 0, 7, 0, 25, 0, 11, 0, 1, 1, 0, 25, 0, 41, 0, 13, 0, 1, 0, 9, 0, 63, 0, 61, 0, 15, 0, 1, 1, 0, 41, 0, 129, 0, 85, 0, 17, 0, 1, 0, 11, 0, 129, 0, 231, 0, 113, 0, 19, 0, 1, 1, 0, 61, 0, 321, 0, 377, 0, 145, 0, 21, 0, 1

Offset: 0

Author

Paul Barry, Sep 17 2005

Keywords

• easy
• nonn
• tabl

Comments

Diagonal sums are aerated Pell numbers.

Examples

			Rows begin
  1;
  0,  1;
  1,  0,  1;
  0,  3,  0,  1;
  1,  0,  5,  0,  1;
  0,  5,  0,  7,  0,  1;
  1,  0, 13,  0,  9,  0,  1;
  0,  7,  0, 25,  0, 11,  0,  1;
  1,  0, 25,  0, 41,  0, 13,  0,  1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000073, A008288, A114123, A216182.

Programs

Magma

function T(n,k)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 0 then return (1+(-1)^n)/2;
  else return T(n-1,k-1) + T(n-2,k) + T(n-3,k-1);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Nov 20 2021

Mathematica

A008288[n_, k_]:= Hypergeometric2F1[-n, -k, 1, 2];
T[n_, k_]:= T[n, k]= (1+(-1)^(n-k))*A008288[(n-k)/2, k]/2;
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2021 *)

Sage

def A008288(n, k): return simplify( hypergeometric([-n, -k], [1], 2) )
def A112743(n, k): return (1 + (-1)^(n-k))*A008288((n-k)/2, k)/2
flatten([[A112743(n,k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Nov 20 2021

Formula

Riordan array (1/(1-x^2), x*(1+x^2)/(1-x^2)).

T(n,k) = Sum_{j=0..k} (1+(-1)^(n-k))*binomial(k,j)*binomial((n-k)/2,j)*2^(j-1).

Sum_{k=0..n} T(n, k) = A000073(n).

T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-3,k-1). - Philippe Deléham, Mar 11 2013