OFFSET
0,5
COMMENTS
Note that the row sums in the example yield the terms of Fibonacci's sequence(A000045). Were the child capable of taking three steps at a time, the row sums of the resulting table would add to the tribonacci sequence (A000073) etc.
Essentially the same as A030528 (without the 0's), where one can find additional information. - Emeric Deutsch, Mar 29 2005
Triangle T(n,k), with zeros omitted, given by (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 08 2012
Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014
REFERENCES
Massimo Nocentini, "An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation", PhD Thesis, University of Florence, 2019. See Ex. 14.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..5775
H.-H. Chern, H.-K. Hwang, T.-H. Tsai, Random unfriendly seating arrangement in a dining table, arXiv preprint arXiv:1406.0614 [math.PR], 2014.
T. Copeland, Addendum to Elliptic Lie Triad
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial
FORMULA
T(n,k) = abs(A092865(n,k)).
O.g.f.: 1/(1-y*x-y*x^2). - Geoffrey Critzer, Dec 27 2011.
EXAMPLE
There are 13 ways for the child to climb a staircase with six steps since the partitions of 6 into 1's and 2's are 222, 2211, 21111 and 111111; and these can be permuted in 1 + 6 + 5 + 1 = 13 ways.
The general cases can be readily shown by displacing Pascal's Triangle (A007318) as follows:
1
..1
..1..1
.....2..1
.....1..3..1
........3..4..1
........1..6..5..1
Triangle (0, 1, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
1
0, 1
0, 1, 1
0, 0, 2, 1
0, 0, 1, 3, 1
0, 0, 0, 3, 4, 1
0, 0, 0, 1, 6, 5, 1 - Philippe Deléham, Feb 08 2012
MAPLE
T:=(n, k)->sum((-1)^(n+i)*binomial(n, i)*binomial(i+k+1, 2*k+1), i=0..n): 1, 1, seq(seq(T(n, k), k=floor(n/2)..n), n=1..16); # Emeric Deutsch, Mar 29 2005
MATHEMATICA
nn = 15; f[list_] := Select[list, # > 0 &];
Map[f, CoefficientList[Series[1/(1 - y x - y x^2), {x, 0, nn}], {x, y}]] // Flatten (* Geoffrey Critzer, Dec 27 2011*)
Table[ Select[ CoefficientList[ Fibonacci[n, x], x], 0 < # &], {n, 0, 17}] // Flatten (* Robert G. Wilson v, May 03 2017 *)
CROSSREFS
KEYWORD
easy,nonn,tabf
AUTHOR
Alford Arnold, Oct 19 2004
EXTENSIONS
More terms from Emeric Deutsch, Mar 29 2005
STATUS
approved