A083859 Main diagonal of generalized Fibonacci array A083856.

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

Author

Paul Barry, May 06 2003

Keywords

• easy
• nonn

Comments

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]

Links

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

Crossrefs

Cf. A083856, A083860, A193376.

Programs

GAP

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

Magma

[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

Maple

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

Mathematica

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 *)

PARI

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

Sage

[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

Formula

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)