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A112743 An aerated Delannoy triangle. 1

%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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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)