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%I #77 Dec 27 2019 11:16:57
%S 0,0,1,0,1,1,0,1,1,1,0,1,1,2,1,0,1,1,3,3,1,0,1,1,4,5,5,1,0,1,1,5,7,11,
%T 8,1,0,1,1,6,9,19,21,13,1,0,1,1,7,11,29,40,43,21,1,0,1,1,8,13,41,65,
%U 97,85,34,1,0,1,1,9,15,55,96,181,217,171,55,1
%N Square array T(n,k) of generalized Fibonacci numbers, read by antidiagonals upwards (n, k >= 0).
%C Row n >= 0 of the array gives the solution to the recurrence b(k) = b(k-1) + n*b(k-2) for k >= 2 with b(0) = 0 and b(1) = 1.
%H A. G. Shannon and J. V. Leyendekkers, <a href="http://nntdm.net/volume-21-2015/number-2/35-42/">The Golden Ratio family and the Binet equation</a>, Notes on Number Theory and Discrete Mathematics, 21(2) (2015), 35-42.
%F T(n, k) = (((1 + sqrt(4*n + 1))/2)^k - ((1 - sqrt(4*n + 1))/2)^k)/sqrt(4*n + 1). [corrected by _Michel Marcus_, Jun 25 2018]
%F From _Thomas Baruchel_, Jun 25 2018: (Start)
%F The g.f. for row n >= 0 is x/(1 - x - n*x^2).
%F The g.f. for column k >= 1 is g(k,x) = 1/(1-x) + Sum_{m = 1..floor((k-1)/2)} (1 - x)^(-1 - m) * binomial(k - 1 - m, m) * Sum_{i = 0..m} x^i * Sum_{j = 0..i} (-1)^j * (i - j)^m * binomial(1 + m, j).
%F The g.f. for column k >= 1 is also g(k,x) = 1 + Sum_{m = 1..floor((k+1)/2)} ((1 - x)^(-m) * binomial(k-m, m-1) * Sum_{j = 0..m} (-1)^j * binomial(m, j) * x^m * Phi(x, -m+1, -j+m)) + Sum_{s = 0..floor((k-1)/2)} binomial(k-s-1, s) * PolyLog(-s, x), where Phi is the Lerch transcendent function. (End)
%F T(n,k) = Sum_{i = 0..k} (-1)^(k+i) * binomial(k,i) * A083857(n,i). - _Petros Hadjicostas_, Dec 24 2019
%e Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... [A057427]
%e 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... [A000045]
%e 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, ... [A001045]
%e 0, 1, 1, 4, 7, 19, 40, 97, 217, 508, ... [A006130]
%e 0, 1, 1, 5, 9, 29, 65, 181, 441, 1165, ... [A006131]
%e 0, 1, 1, 6, 11, 41, 96, 301, 781, 2286, ... [A015440]
%e 0, 1, 1, 7, 13, 55, 133, 463, 1261, 4039, ... [A015441]
%e 0, 1, 1, 8, 15, 71, 176, 673, 1905, 6616, ... [A015442]
%e 0, 1, 1, 9, 17, 89, 225, 937, 2737, 10233, ... [A015443]
%e 0, 1, 1, 10, 19, 109, 280, 1261, 3781, 15130, ... [A015445]
%e ...
%p A083856_row := proc(r, n) local R; R := proc(n) option remember;
%p if n<=1 then n else R(n-1)+r*R(n-2) fi end: R(n) end:
%p for r from 0 to 9 do seq(A083856_row(r, n), n=0..9) od; # _Peter Luschny_, Mar 06 2017
%t T[_, 0] = 0; T[_, 1|2] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + n T[n, k-2];
%t Table[T[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%o (Julia)
%o function generalized_fibonacci(r, n)
%o F = BigInt[1 r; 1 0]
%o Fn = F^n
%o Fn[2, 1]
%o end
%o for r in 0:6 println([generalized_fibonacci(r, n) for n in 0:9]) end # _Peter Luschny_, Mar 06 2017
%Y Rows include A057427 (n=0), A000045 (n=1), A001045 (n=2), A006130 (n=3), A006131 (n=4), A015440 (n=5), A015441 (n=6), A015442 (n=7), A015443 (n=8), A015445 (n=9).
%Y Columns include A000012 (k=1,2), A000027 (k=3), A005408 (k=4), A028387 (k=5), A000567 (k=6), A106734 (k=7).
%Y Cf. A083857 (binomial transform), A083859 (main diagonal), A083860 (first subdiagonal), A083861 (second binomial transform), A110112, A110113 (diagonal sums), A193376 (transposed variant), A172237 (transposed variant).
%K nonn,tabl,easy
%O 0,14
%A _Paul Barry_, May 06 2003
%E Various sections edited by _Petros Hadjicostas_, Dec 24 2019
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