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

0,5

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

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

Sequence in context: A287601 A035667 A092865 * A102426 A052920 A320250

Adjacent sequences: A098922 A098923 A098924 * A098926 A098927 A098928

Keyword

easy,nonn,tabf

Author

Alford Arnold, Oct 19 2004

Extensions

More terms from Emeric Deutsch, Mar 29 2005

Status

approved