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A112743 An aerated Delannoy triangle.

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