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A073134 Table by antidiagonals of T(n,k)=n*T(n,k-1)-T(n,k-2) starting with T(n,1)=1.

%I A073134 #18 Jan 05 2025 19:51:37
%S A073134 1,1,1,0,2,1,-1,3,3,1,-1,4,8,4,1,0,5,21,15,5,1,1,6,55,56,24,6,1,1,7,
%T A073134 144,209,115,35,7,1,0,8,377,780,551,204,48,8,1,-1,9,987,2911,2640,
%U A073134 1189,329,63,9,1,-1,10,2584,10864,12649,6930,2255,496,80,10,1,0,11,6765,40545,60605,40391,15456,3905,711,99,11,1,1,12
%N A073134 Table by antidiagonals of T(n,k)=n*T(n,k-1)-T(n,k-2) starting with T(n,1)=1.
%H A073134 Shmuel T. Klein, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/29-2/klein.pdf">Combinatorial Representation of Generalized Fibonacci Numbers</a>, Fib. Quarterly 29 (2) (1991) 124-131, variable U_n^m. [From _R. J. Mathar_, Feb 19 2010]
%F A073134 T(n, k) = A073133(n, k)-2*A073135(n, k-2).
%F A073134 T(n, k) = Sum_{j=0..k-1} A049310(k-1, j)*n^j.
%e A073134 Rows start:
%e A073134   1, 1,  0, -1,  -1,   0,    1, ...;
%e A073134   1, 2,  3,  4,   5,   6,    7, ...;
%e A073134   1, 3,  8, 21,  55, 144,  377, ...;
%e A073134   1, 4, 15, 56, 209, 780, 2911, ...;
%e A073134   ...
%o A073134 (PARI) T(n,k) = sum(j=0,k-1,A049310(k-1,j)*n^j) \\ _Jason Yuen_, Aug 20 2024
%Y A073134 Rows include A010892, A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190. Columns include (with some gaps) A000012, A000027, A005563, A057722.
%Y A073134 Cf. A094954.
%K A073134 sign,tabl
%O A073134 1,5
%A A073134 _Henry Bottomley_, Jul 16 2002