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A060964 Table by antidiagonals where T(n,k) = n*T(n,k-1) - T(n,k-2) with T(n,0) = 2 and T(n,1) = n.

%I A060964 #13 Sep 08 2022 08:45:03
%S A060964 2,0,2,-2,1,2,0,-1,2,2,2,-2,2,3,2,0,-1,2,7,4,2,-2,1,2,18,14,5,2,0,2,2,
%T A060964 47,52,23,6,2,2,1,2,123,194,110,34,7,2,0,-1,2,322,724,527,198,47,8,2,
%U A060964 -2,-2,2,843,2702,2525,1154,322,62,9,2,0,-1,2,2207,10084,12098,6726,2207,488,79,10,2
%N A060964 Table by antidiagonals where T(n,k) = n*T(n,k-1) - T(n,k-2) with T(n,0) = 2 and T(n,1) = n.
%H A060964 G. C. Greubel, <a href="/A060964/b060964.txt">Antidiagonals n = 0..100, flattened</a>
%F A060964 For all m, T(n, k) = T(n, |m|)*T(n, |k - m|) - T(n, |k - 2m|).
%F A060964 T(n, 2k) = T(n, k)^2 - 2.
%F A060964 T(n, 2k + 1) = T(n, k)*T(n, k + 1) - n.
%F A060964 T(n, 3k) = T(n, k)^3 - 3*T(n, k).
%F A060964 T(n, 4k) = T(n, k)^4 - 4*T(n, k)^2 + 2.
%F A060964 T(n, 5k) = T(n, k)^5 - 5*T(n, k)^3 + 5*T(n, k) etc.
%F A060964 T(n, -k) = T(n, k).
%F A060964 T(-n, k) = T(-n, -k) = (-1)^k * T(n, k).
%F A060964 T(n, k) = ( n*( ((n + sqrt(n^2 -4))/2)^k - ((n - sqrt(n^2 -4))/2)^k ) - 2*( ((n + sqrt(n^2 -4))/2)^(k-1) - ((n - sqrt(n^2 -4))/2)^(k-1) ) )/sqrt(n^2 -4).
%F A060964 T(n, k) = n*ChebyshevU(k-1, n/2) - 2*ChebyshevU(k-2, n/2). - _G. C. Greubel_, Jan 15 2020
%e A060964 Square array begins as:
%e A060964   2, 0, -2,   0,   2,    0,    -2, ...
%e A060964   2, 1, -1,  -2,  -1,    1,     2, ...
%e A060964   2, 2,  2,   2,   2,    2,     2, ...
%e A060964   2, 3,  7,  18,  47,  123,   322, ...
%e A060964   2, 4, 14,  52, 194,  724,  2702, ...
%e A060964   2, 5, 23, 110, 527, 2525, 12098, ...
%p A060964 seq(seq( simplify(k*ChebyshevU(n-k-1, k/2) -2*ChebyshevU(n-k-2, k/2)), k=0..n), n=0..12); # _G. C. Greubel_, Jan 15 2020
%t A060964 Table[k*ChebyshevU[n-k-1, k/2] -2*ChebyshevU[n-k-2, k/2], {n,0,12}, {k,0,n} ]//Flatten
%o A060964 (PARI) T(n,k) = n*polchebyshev(k-1,2,n/2) -2*polchebyshev(k-2,2,n/2);
%o A060964 for(n=0,12, for(k=0,n, print1(T(k,n-k), ", "))) \\ _G. C. Greubel_, Jan 15 2020
%o A060964 (Magma)
%o A060964 function T(n,k)
%o A060964   if k eq 0 then return 2;
%o A060964   elif k eq 1 then return n;
%o A060964   else return n*T(n, k-1) - T(n, k-2);
%o A060964   end if; return T; end function;
%o A060964 [T(k,n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 15 2020
%o A060964 (Sage) [[k*chebyshev_U(n-k-1, k/2) -2*chebyshev_U(n-k-2, k/2) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 15 2020
%o A060964 (GAP)
%o A060964 T:= function(n,k)
%o A060964     if k=0 then return 2;
%o A060964     elif k=1 then return n;
%o A060964     else return n*T(n,k-1) - T(n,k-2);
%o A060964     fi; end;
%o A060964 Flat(List([0..12], n-> List([0..n], k-> T(k,n-k) ))); # _G. C. Greubel_, Jan 15 2020
%Y A060964 Rows include A003499, A003500, A003501, A005248, A007395, A056854, A056918, A057079.
%Y A060964 Columns include A001477, A007395, A008865, A058794.
%K A060964 sign,tabl
%O A060964 0,1
%A A060964 _Henry Bottomley_, May 09 2001