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A060959 Table by antidiagonals of generalized Fibonacci numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=0 and T(n,1)=1.

%I A060959 #8 Sep 08 2022 08:45:03
%S A060959 0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,3,3,1,1,0,1,5,5,4,1,1,0,1,8,11,7,5,1,
%T A060959 1,0,1,13,21,19,9,6,1,1,0,1,21,43,40,29,11,7,1,1,0,1,34,85,97,65,41,
%U A060959 13,8,1,1,0,1,55,171,217,181,96,55,15,9,1,1,0,1,89,341,508,441,301,133,71,17,10,1,1,0
%N A060959 Table by antidiagonals of generalized Fibonacci numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=0 and T(n,1)=1.
%H A060959 G. C. Greubel, <a href="/A060959/b060959.txt">Antidiagonals n = 0..100, flattened</a>
%F A060959 T(n, k) = ( ((1+sqrt(1+4*n))/2)^k - ((1-sqrt(1+4*n))/2)^k )/sqrt(1+4*n).
%e A060959 Square array begins as:
%e A060959   0, 1, 1, 1,  1,  1,  1, ...
%e A060959   0, 1, 1, 2,  3,  5,  8, ...
%e A060959   0, 1, 1, 3,  5, 11, 21, ...
%e A060959   0, 1, 1, 4,  7, 19, 40, ...
%e A060959   0, 1, 1, 5,  9, 29, 65, ...
%e A060959   0, 1, 1, 6, 11, 41, 96, ...
%p A060959 seq(seq( round((((1+sqrt(1+4*k))/2)^(n-k) - ((1-sqrt(1+4*k))/2)^(n-k) )/sqrt(1+4*k)), k=0..n), n=0..12); # _G. C. Greubel_, Jan 15 2020
%t A060959 T[n_, k_]:= If[n==k==0, 0, Round[(((1+Sqrt[1+4n])/2)^k - ((1-Sqrt[1+4n])/2)^k)/Sqrt[1+4n]]]; Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 15 2020 *)
%o A060959 (PARI) T(n,k) = ( ((1+sqrt(1+4*n))/2)^k - ((1-sqrt(1+4*n))/2)^k )/sqrt(1+4*n);
%o A060959 for(n=0,12, for(k=0,n, print1( round(T(k,n-k)), ", "))) \\ _G. C. Greubel_, Jan 15 2020
%o A060959 (Magma) [Round( (((1+Sqrt(1+4*k))/2)^(n-k) - ((1-Sqrt(1+4*k))/2)^(n-k) )/Sqrt(1+4*k) ): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 15 2020
%o A060959 (Sage) [[ round( (((1+sqrt(1+4*k))/2)^(n-k) - ((1-sqrt(1+4*k))/2)^(n-k) )/sqrt(1+4*k) ) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 15 2020
%o A060959 (GAP) Flat(List([0..12], n-> List([0..n], k-> (((1+Sqrt(1+4*k))/2)^(n-k) - ((1-Sqrt(1+4*k))/2)^(n-k))/Sqrt(1+4*k) ))); # _G. C. Greubel_, Jan 15 2020
%Y A060959 Rows include A057427, A000045, A001045, A006130, A006131, A015440, A015441, A015442, A015443, A015445, A015446, A015447, A053404.
%Y A060959 Columns include A000004, A000012 (twice), A000027, A000567 A005408, A028387.
%K A060959 nonn,tabl
%O A060959 0,12
%A A060959 _Henry Bottomley_, May 10 2001