%I #17 Sep 08 2022 08:45:23
%S 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,
%T 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,
%U 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
%N An aerated Delannoy triangle.
%C Diagonal sums are aerated Pell numbers.
%H G. C. Greubel, <a href="/A112743/b112743.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Riordan array (1/(1-x^2), x*(1+x^2)/(1-x^2)).
%F T(n,k) = Sum_{j=0..k} (1+(-1)^(n-k))*binomial(k,j)*binomial((n-k)/2,j)*2^(j-1).
%F Sum_{k=0..n} T(n, k) = A000073(n).
%F T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-3,k-1). - _Philippe Deléham_, Mar 11 2013
%e Rows begin
%e 1;
%e 0, 1;
%e 1, 0, 1;
%e 0, 3, 0, 1;
%e 1, 0, 5, 0, 1;
%e 0, 5, 0, 7, 0, 1;
%e 1, 0, 13, 0, 9, 0, 1;
%e 0, 7, 0, 25, 0, 11, 0, 1;
%e 1, 0, 25, 0, 41, 0, 13, 0, 1;
%t A008288[n_, k_]:= Hypergeometric2F1[-n, -k, 1, 2];
%t T[n_, k_]:= T[n, k]= (1+(-1)^(n-k))*A008288[(n-k)/2, k]/2;
%t Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 20 2021 *)
%o (Magma)
%o function T(n,k)
%o if k lt 0 or k gt n then return 0;
%o elif k eq n then return 1;
%o elif k eq 0 then return (1+(-1)^n)/2;
%o else return T(n-1,k-1) + T(n-2,k) + T(n-3,k-1);
%o end if;
%o return T;
%o end function;
%o [T(n,k): k in [0..n], n in [0..14]]; // _G. C. Greubel_, Nov 20 2021
%o (Sage)
%o def A008288(n, k): return simplify( hypergeometric([-n, -k], [1], 2) )
%o def A112743(n, k): return (1 + (-1)^(n-k))*A008288((n-k)/2, k)/2
%o flatten([[A112743(n,k) for k in (0..n)] for n in (0..14)]) # _G. C. Greubel_, Nov 20 2021
%Y Cf. A000073, A008288, A114123, A216182.
%K easy,nonn,tabl
%O 0,8
%A _Paul Barry_, Sep 17 2005
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