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

0,8

Comments

Diagonal sums are aerated Pell numbers.

Links

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

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A000073, A008288, A114123, A216182.

Sequence in context: A317595 A263753 A303877 * A230427 A229995 A119467

Adjacent sequences: A112740 A112741 A112742 * A112744 A112745 A112746

Keyword

easy,nonn,tabl

Author

Paul Barry, Sep 17 2005

Status

approved