A110113 -id:A110113 - cp's OEIS frontend

A083856 Square array T(n,k) of generalized Fibonacci numbers, read by antidiagonals upwards (n, k >= 0).

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, 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, 97, 85, 34, 1, 0, 1, 1, 9, 15, 55, 96, 181, 217, 171, 55, 1

Offset: 0

Paul Barry, May 06 2003

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.

			Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  0, 1, 1,  1,  1,   1,   1,    1,    1,     1, ... [A057427]
  0, 1, 1,  2,  3,   5,   8,   13,   21,    34, ... [A000045]
  0, 1, 1,  3,  5,  11,  21,   43,   85,   171, ... [A001045]
  0, 1, 1,  4,  7,  19,  40,   97,  217,   508, ... [A006130]
  0, 1, 1,  5,  9,  29,  65,  181,  441,  1165, ... [A006131]
  0, 1, 1,  6, 11,  41,  96,  301,  781,  2286, ... [A015440]
  0, 1, 1,  7, 13,  55, 133,  463, 1261,  4039, ... [A015441]
  0, 1, 1,  8, 15,  71, 176,  673, 1905,  6616, ... [A015442]
  0, 1, 1,  9, 17,  89, 225,  937, 2737, 10233, ... [A015443]
  0, 1, 1, 10, 19, 109, 280, 1261, 3781, 15130, ... [A015445]
  ...

A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, 21(2) (2015), 35-42.

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).
Columns include A000012 (k=1,2), A000027 (k=3), A005408 (k=4), A028387 (k=5), A000567 (k=6), A106734 (k=7).
Cf. A083857 (binomial transform), A083859 (main diagonal), A083860 (first subdiagonal), A083861 (second binomial transform), A110112, A110113 (diagonal sums), A193376 (transposed variant), A172237 (transposed variant).

function generalized_fibonacci(r, n)
   F = BigInt[1 r; 1 0]
   Fn = F^n
   Fn[2, 1]
end
for r in 0:6 println([generalized_fibonacci(r, n) for n in 0:9]) end # Peter Luschny, Mar 06 2017

A083856_row := proc(r, n) local R; R := proc(n) option remember;
if n<=1 then n else R(n-1)+r*R(n-2) fi end: R(n) end:
for r from 0 to 9 do seq(A083856_row(r, n), n=0..9) od; # Peter Luschny, Mar 06 2017

T[, 0] = 0; T[, 1|2] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + n T[n, k-2];
Table[T[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

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]
From Thomas Baruchel, Jun 25 2018: (Start)
The g.f. for row n >= 0 is x/(1 - x - n*x^2).
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).
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)
T(n,k) = Sum_{i = 0..k} (-1)^(k+i) * binomial(k,i) * A083857(n,i). - Petros Hadjicostas, Dec 24 2019

Various sections edited by Petros Hadjicostas, Dec 24 2019

A172237 T(n,k) = T(n-1,k) + k*T(n-2,k) for k >= 1 and n >= 3 with T(0,k) = 0 and T(1,k) = T(2,k) = 1 for all k >= 1; array T(n,k), read by descending antidiagonals, with n >= 0 and k >= 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 3, 0, 1, 1, 4, 5, 5, 0, 1, 1, 5, 7, 11, 8, 0, 1, 1, 6, 9, 19, 21, 13, 0, 1, 1, 7, 11, 29, 40, 43, 21, 0, 1, 1, 8, 13, 41, 65, 97, 85, 34, 0, 1, 1, 9, 15, 55, 96, 181, 217, 171, 55, 0, 1, 1, 10, 17, 71, 133, 301, 441, 508, 341, 89, 0, 1, 1, 11

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 29 2010

Transposed variant of A083856, without the top row of A083856.
Antidiagonal sums are (0, 1, 2, 4, 8, 16, 33, 70, 153, 345, ...) = (A110113(n) - 1: n >= 1).
Characteristic polynomials for columns are y^2 - y - k.

			Array T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
    0,    0,    0,    0,    0,     0,     0,     0,     0,     0, ...
    1,    1,    1,    1,    1,     1,     1,     1,     1,     1, ...
    1,    1,    1,    1,    1,     1,     1,     1,     1,     1, ...
    2,    3,    4,    5,    6,     7,     8,     9,    10,    11, ...
    3,    5,    7,    9,   11,    13,    15,    17,    19,    21, ...
    5,   11,   19,   29,   41,    55,    71,    89,   109,   131, ...
    8,   21,   40,   65,   96,   133,   176,   225,   280,   341, ...
   13,   43,   97,  181,  301,   463,   673,   937,  1261,  1651, ...
   21,   85,  217,  441,  781,  1261,  1905,  2737,  3781,  5061, ...
   34,  171,  508, 1165, 2286,  4039,  6616, 10233, 15130, 21571, ...
   55,  341, 1159, 2929, 6191, 11605, 19951, 32129, 49159, 72181, ...
   ...

Cf. A083856, A110113, A193376.

A172237 := proc(n,k)
        if n = 0 then
                0;
        elif n <=2 then
                1 ;
        else
                procname(n-1,k)+k*procname(n-2,k) ;
        end if;
end proc: # R. J. Mathar, Jul 05 2012

f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
m1 = Table[f[n, a], {n, 0, 10}, {a, 1, 11}];
Table[Table[m1[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]

More terms from Petros Hadjicostas, Dec 26 2019

cp's OEIS Frontend

A083856 Square array T(n,k) of generalized Fibonacci numbers, read by antidiagonals upwards (n, k >= 0).

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