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A098925
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Distribution of the number of ways for a child to climb a staircase having r steps (one step or two steps at a time).
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6
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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; internal format)
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OFFSET
| 0,5
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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 (deutsch(AT)duke.poly.edu), 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. - DELEHAM Philippe, Feb 08 2012
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FORMULA
| a(n) = abs(A092865)
O.g.f.: 1/(1-yx-yx^2) -Geoffrey Critzer, Dec 27 2011.
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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 - DELEHAM Philippe, Feb 08 2012
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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); (Deutsch)
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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*)
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CROSSREFS
| Cf. A000045, A000073, A000078, A007318, A092865.
Cf. 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: A035667 A102426 A092865 * A052920 A089141 A170820
Adjacent sequences: A098922 A098923 A098924 * A098926 A098927 A098928
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Alford Arnold (Alford1940(AT)aol.com), Oct 19 2004
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2005
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