A098925 Distribution of the number of ways for a child to climb a staircase having r steps (one step or two steps at a time).

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, 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, 1, 28, 126, 210, 165, 66, 13, 1, 8, 84, 252, 330, 220, 78, 14, 1, 1, 36, 210, 462, 495, 286, 91

Offset: 0

Author

Alford Arnold, Oct 19 2004

Keywords

• easy
• nonn
• tabf

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

Examples

			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
		

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

Crossrefs

Cf. A000045, A000073, A007318, A092865, A030528.

All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - N. J. A. Sloane, May 29 2011

Programs

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 *)

Formula

T(n,k) = abs(A092865(n,k)).

O.g.f.: 1/(1-y*x-y*x^2). - Geoffrey Critzer, Dec 27 2011.

Extensions

More terms from Emeric Deutsch, Mar 29 2005