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A083857 Square array T(n,k) of binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.

%I A083857 #47 Dec 30 2019 18:57:03
%S A083857 0,0,1,0,1,3,0,1,3,7,0,1,3,8,15,0,1,3,9,21,31,0,1,3,10,27,55,63,0,1,3,
%T A083857 11,33,81,144,127,0,1,3,12,39,109,243,377,255,0,1,3,13,45,139,360,729,
%U A083857 987,511,0,1,3,14,51,171,495,1189,2187,2584,1023,0,1,3,15,57,205,648
%N A083857 Square array T(n,k) of binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.
%C A083857 Row n >= 0 of the array gives the solution to the recurrence b(k) = 3*b(k-1) + (n-2) * a(k-2) for k >= 2 with a(0) = 0 and a(1) = 1. These are the binomial transforms of the rows of the generalized Fibonacci numbers A083856.
%H A083857 OEIS, <a href="https://oeis.org/transforms.html">Transformations of integer sequences</a>.
%F A083857 T(n, k) = ((3 + sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) - ((3 - sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) for n, k >= 0.
%F A083857 O.g.f. of row n >= 0: -x/(-1 + 3*x + (n-2)*x^2) . - _R. J. Mathar_, Nov 23 2007
%F A083857 T(n,k) = Sum_{i = 0..k} binomial(k,i)*A083856(n,i). - _Petros Hadjicostas_, Dec 24 2019
%e A083857 Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e A083857   0, 1, 3,  7, 15,  31,  63,  127,  255, ...
%e A083857   0, 1, 3,  8, 21,  55, 144,  377,  987, ...
%e A083857   0, 1, 3,  9, 27,  81, 243,  729, 2187, ...
%e A083857   0, 1, 3, 10, 33, 109, 360, 1189, 3927, ...
%e A083857   0, 1, 3, 11, 39, 139, 495, 1763, 6279, ...
%e A083857   0, 1, 3, 12, 45, 171, 648, 2457, 9315, ...
%e A083857   ...
%Y A083857 Rows include A000225 (n=0), A001906 (n=1), A000244 (n=2), A006190 (n=3), A007482 (n=4), A030195 (n=5), A015521 (n=6).
%Y A083857 Cf. A083856, A083861 (binomial transform), A083862 (main diagonal).
%K A083857 easy,nonn,tabl
%O A083857 0,6
%A A083857 _Paul Barry_, May 06 2003
%E A083857 Various sections edited by _Petros Hadjicostas_, Dec 24 2019