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%I #66 Dec 03 2018 21:08:06
%S 1,1,1,1,2,1,1,3,1,3,4,1,1,6,5,1,4,10,6,1,1,10,15,7,1,5,20,21,8,1,1,
%T 15,35,28,9,1,6,35,56,36,10,1,1,21,70,84,45,11,1,7,56,126,120,55,12,1,
%U 1,28,126,210,165,66,13,1,8,84,252,330,220,78,14,1,1,36,210,462,495,286,91
%N Distribution of the number of ways for a child to climb a staircase having r steps (one step or two steps at a time).
%C 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.
%C Essentially the same as A030528 (without the 0's), where one can find additional information. - _Emeric Deutsch_, Mar 29 2005
%C 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
%C 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
%D 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.
%H Vincenzo Librandi, <a href="/A098925/b098925.txt">Table of n, a(n) for n = 0..5775</a>
%H H.-H. Chern, H.-K. Hwang, T.-H. Tsai, <a href="http://arxiv.org/abs/1406.0614">Random unfriendly seating arrangement in a dining table</a>, arXiv preprint arXiv:1406.0614 [math.PR], 2014.
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>
%H P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>
%F T(n,k) = abs(A092865(n,k)).
%F O.g.f.: 1/(1-y*x-y*x^2). - _Geoffrey Critzer_, Dec 27 2011.
%e 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.
%e The general cases can be readily shown by displacing Pascal's Triangle (A007318) as follows:
%e 1
%e ..1
%e ..1..1
%e .....2..1
%e .....1..3..1
%e ........3..4..1
%e ........1..6..5..1
%e Triangle (0, 1, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
%e 1
%e 0, 1
%e 0, 1, 1
%e 0, 0, 2, 1
%e 0, 0, 1, 3, 1
%e 0, 0, 0, 3, 4, 1
%e 0, 0, 0, 1, 6, 5, 1 - _Philippe Deléham_, Feb 08 2012
%p 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
%t nn = 15; f[list_] := Select[list, # > 0 &];
%t Map[f, CoefficientList[Series[1/(1 - y x - y x^2), {x, 0, nn}], {x, y}]] // Flatten (* _Geoffrey Critzer_, Dec 27 2011*)
%t Table[ Select[ CoefficientList[ Fibonacci[n, x], x], 0 < # &], {n, 0, 17}] // Flatten (* _Robert G. Wilson v_, May 03 2017 *)
%Y Cf. A000045, A000073, A007318, A092865, A030528.
%Y All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - _N. J. A. Sloane_, May 29 2011
%K easy,nonn,tabf
%O 0,5
%A _Alford Arnold_, Oct 19 2004
%E More terms from _Emeric Deutsch_, Mar 29 2005
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