A193376 -id:A193376 - 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

A171180 a(n) = (4n + 1)^(1/2)/(4n + 1)((1 - p)q^n - (1 - q)p^n), where p = (1 - (4n + 1)^(1/2))/2 and q = (1 + (4*n + 1)^(1/2))/2.

Original entry on oeis.org

1, 3, 7, 29, 96, 463, 1905, 10233, 49159, 287891, 1557744, 9814741, 58451849, 392539575, 2532516511, 17999936497, 124360077816, 930257069563, 6822980957481, 53470578301581, 413527226164711, 3382254701784223, 27432377661111360, 233410016529114601

Offset: 1

Gary Detlefs, Dec 04 2009

nonn

If a sequence (s(n): n >= 0) is of the form s(0) = x, s(1) = x, and s(n) = s(n-1) + k*s(n-2) for n >= 2 (for some integer k >= 1 and some number x), then s(k) = a(k)*x. For example, if k = 6 and x = 3, then (s(n): n = 0..6) = (3, 3, 21, 39, 165, 399, 1389) and s(6) = 1389 = 463*3 = a(6)*x. [Edited by Petros Hadjicostas, Dec 26 2019]

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.

Cf. A350467.

Table[Sum[Binomial[n - k, k]*n^k, {k, 0, n}], {n, 1, 25}] (* Vaclav Kotesovec, Jan 08 2024 *)
Table[Hypergeometric2F1[(1 - n)/2, -n/2, -n, -4*n], {n, 1, 25}] (* Vaclav Kotesovec, Jan 08 2024 *)

{a(n)=polcoeff(1/(1-x-n*x^2+x*O(x^n)), n)} \\ Paul D. Hanna, Dec 27 2012

a(n) = A193376(n,n). - Olivier Gérard, Jul 25 2011
a(n) = [x^n] 1/(1 - x - n*x^2). - Paul D. Hanna, Dec 27 2012
From Vaclav Kotesovec, Jan 08 2024: (Start)
a(n) = Sum_{k=0..n} binomial(n-k,k) * n^k.
a(n) ~ exp(sqrt(n)/2) * n^(n/2) / 2 * (1 + 23/(48*sqrt(n))). (End)

A083859 Main diagonal of generalized Fibonacci array A083856.

Original entry on oeis.org

0, 1, 1, 4, 9, 41, 133, 673, 2737, 15130, 72181, 430739, 2320825, 14815529, 88005541, 596681296, 3843559137, 27515587661, 189933449365, 1428716457761, 10474213334761, 82448447397646, 637534807917701, 5233087759204967, 42445677865505425, 362213650380301201

Offset: 0

Paul Barry, May 06 2003

If a sequence (s(n): n >= 0) is of the form s(0) = 0, s(1) = x, and s(n) = s(n-1) + k*s(n-2) for n >= 2 (for some integer k >= 0 and some number x), then s(k) = a(k)*x. For example if k = 7 and x = 5, then (s(n): n = 0..7) = (0, 5, 5, 40, 75, 355, 880, 3365) and s(7) = 3365 = 673*5 = a(7)*x. - Gary Detlefs, Dec 04 2009 [Edited by Petros Hadjicostas, Dec 24 2019]

G. C. Greubel, Table of n, a(n) for n = 0..690

Cf. A083856, A083860, A193376.

Concatenation([0], List([1..30], n-> Sum([0..Int((n-1)/2)], j-> Binomial(n-j-1, j)*n^j) )); # G. C. Greubel, Dec 27 2019

[0] cat [ &+[Binomial(n-j-1, j)*n^j: j in [0..Floor((n-1)/2)]] : n in [1..30]]; // G. C. Greubel, Dec 27 2019

seq( `if`(n=0, 0, simplify( (-sqrt(n)*I)^(n-1)*ChebyshevU(n-1, I/(2*sqrt(n)))) ), n=0..30); # G. C. Greubel, Dec 27 2019
# second Maple program:
a:= n-> (<<0|1>, >^n)[1, 2]:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 19 2021

Table[DifferenceRoot[Function[{y, m}, {y[2 + m] == y[1 + m] + n*y[m], y[0] == 0, y[1] == 1}]][n], {n, 0, 20}] (* Benedict W. J. Irwin, Nov 03 2016 *)
Table[If[n==0, 0, Round[(Sqrt[n])^(n-1)*Fibonacci[n, 1/Sqrt[n]] ]], {n,0,30}] (* G. C. Greubel, Dec 27 2019 *)

vector(31, n, if(n==1, 0, round((-sqrt(n-1)*I)^(n-2)*polchebyshev(n-2, 2, I/(2*sqrt(n-1)))) ) ) \\ G. C. Greubel, Dec 27 2019

[0]+[(-sqrt(n)*I)^(n-1)*chebyshev_U(n-1, I/(2*sqrt(n))) for n in (1..30)] # G. C. Greubel, Dec 27 2019

a(n) = (((1 + sqrt(4*n + 1))/2)^n - ((1 - sqrt(4*n + 1))/2)^n)/sqrt(4*n + 1).
a(n) = A193376(n-1,n) for n >= 2. - R. J. Mathar, Aug 23 2011
a(n) = y(n,n), where y(m+2,n) = y(m+1,n) + n*y(m,n) with y(0,n) = 0 and y(1,n) = 1 for all n. - Benedict W. J. Irwin, Nov 03 2016
a(n) = [x^n] x/(1 - x - n*x^2). - Ilya Gutkovskiy, Oct 10 2017
a(n) = Sum_{s = 0..floor((n-1)/2)} binomial(n-1-s, s) * n^s. - Petros Hadjicostas, Dec 24 2019
From G. C. Greubel, Dec 27 2019: (Start)
a(n) = (sqrt(n))^n * Fibonacci(n, 1/sqrt(n)), with a(0)=0.
a(n) = (-sqrt(n)*i)^(n-1)*ChebyshevU(n-1, i/(2*sqrt(n))), with a(0)=0. (End)

A083860 First subdiagonal of generalized Fibonacci array A083856.

Original entry on oeis.org

0, 1, 1, 5, 11, 55, 176, 937, 3781, 21571, 102455, 624493, 3356640, 21752431, 129055681, 884773585, 5696734715, 41129090011, 283908657880, 2149818248341, 15765656131765, 124759995175751, 965186517474191, 7956847444317049, 64577172850366176, 553048437381116275

Offset: 0

Paul Barry, May 06 2003

Cf. A083856, A083859.

T := proc(n, k) local v; option remember; if 0 <= n and k = 0 then v := 0; end if; if 0 <= n and k = 1 then v := 1; end if; if 0 <= n and 2 <= k then v := T(n, k - 1) + n*T(n, k - 2); end if; v; end proc;
seq(T(n + 1, n), n = 0 .. 40); # Petros Hadjicostas, Dec 25 2019

T[, 0] = 0; T[, 1|2] = 1;
T[n_, k_] := T[n, k] = T[n, k-1] + n T[n, k-2];
a[n_] := T[n+1, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Sep 26 2022 *)

a(n) = (((1 + sqrt(4*n + 5))/2)^n - ((1 - sqrt(4*n + 5))/2)^n)/sqrt(4*n + 5).
a(n) = A193376(n-1, n+1) for n >= 2. - R. J. Mathar, Aug 23 2011
a(n) = Sum_{s = 0..floor((n-1)/2)} binomial(n-1-s, s) * (n+1)^s. - Petros Hadjicostas, Dec 25 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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A171180 a(n) = (4*n + 1)^(1/2)/(4*n + 1)*((1 - p)*q^n - (1 - q)*p^n), where p = (1 - (4*n + 1)^(1/2))/2 and q = (1 + (4*n + 1)^(1/2))/2.

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A083859 Main diagonal of generalized Fibonacci array A083856.

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A083860 First subdiagonal of generalized Fibonacci array A083856.

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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.

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A171180 a(n) = (4n + 1)^(1/2)/(4n + 1)((1 - p)q^n - (1 - q)p^n), where p = (1 - (4n + 1)^(1/2))/2 and q = (1 + (4*n + 1)^(1/2))/2.