A083856 - OEIS

%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