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 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). 12
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 10 23:14 EST 2019. Contains 329909 sequences. (Running on oeis4.)