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A119473
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Triangle read by rows: T(n,k) is number of binary words of length n and having k runs of 0's of odd length, 0 <= k <= ceiling(n/2). (A run of 0's is a subsequence of consecutive 0's of maximal length.)
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4
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1, 1, 1, 2, 2, 3, 4, 1, 5, 8, 3, 8, 15, 8, 1, 13, 28, 19, 4, 21, 51, 42, 13, 1, 34, 92, 89, 36, 5, 55, 164, 182, 91, 19, 1, 89, 290, 363, 216, 60, 6, 144, 509, 709, 489, 170, 26, 1, 233, 888, 1362, 1068, 446, 92, 7, 377, 1541, 2580, 2266, 1105, 288, 34, 1, 610, 2662, 4830
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OFFSET
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0,4
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COMMENTS
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Row n has 1+ceiling(n/2) terms.
T(n,0) = Fibonacci(n+1) = A000045(n+1).
Triangle, with zeros included, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 07 2011
T(n,k) is the number of compositions of n+1 that have exactly k even parts. - Geoffrey Critzer, Mar 03 2012
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REFERENCES
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I. Goulden and D. Jackson, Combinatorial Enumeration, John Wiley and Sons, 1983, page 54.
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LINKS
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Rigoberto Flórez, Javier González, Mateo Matijasevick, Cristhian Pardo, José Luis Ramírez, Lina Simbaqueba, and Fabio Velandia, Lattice paths in corridors and cyclic corridors, Contrib. Disc. Math. (2024) Vol. 19. No. 2, 36-55. See p. 17.
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FORMULA
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G.f.: (1+t*z)/(1-z-z^2-t*z^2).
G.f. of column k (k>=1): z^(2*k-1)*(1-z^2)/(1-z-z^2)^(k+1).
Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A000079(n), A105476(n+1), A159612(n+1), A189732(n+1) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Dec 07 2011
G.f.: (1+x*y)*T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x/((2*k+2+ x*(1+y))*x + 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
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EXAMPLE
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T(5,2)=8 because we have 00010, 01000, 01011, 01101, 01110, 10101, 10110 and 11010.
T(5,2)=8 because there are 8 compositions of 6 that have 2 even parts: 4+2, 2+4, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2. - Geoffrey Critzer, Mar 03 2012
Triangle starts:
1;
1, 1;
2, 2;
3, 4, 1;
5, 8, 3;
8, 15, 8, 1;
Triangle (1,1,-1,0,0,0...) DELTA (1,-1,0,0,0,...) begins:
1;
1, 1;
2, 2, 0;
3, 4, 1, 0;
5, 8, 3, 0, 0;
8, 15, 8, 1, 0, 0;
13, 28, 19, 4, 0, 0, 0;
21, 51, 42, 13, 1, 0, 0, 0;
34, 92, 89, 36, 5, 0, 0, 0, 0; ... (End)
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MAPLE
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G:=(1+t*z)/(1-z-z^2-t*z^2): Gser:=simplify(series(G, z=0, 18)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 0 to 14 do seq(coeff(P[n], t, j), j=0..ceil(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; local j; if n=0 then 1
else []; for j to n do zip((x, y)->x+y, %,
[`if`(irem(j, 2)=0, 0, NULL), b(n-j)], 0) od; %[] fi
end:
T:= n-> b(n+1):
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MATHEMATICA
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f[list_] := Select[list, # > 0 &]; nn = 15; a = (x + y x^2)/(1 - x^2); Map[f, Drop[CoefficientList[Series[1/(1 - a), {x, 0, nn}], {x, y}], 1]] // Flatten (* Geoffrey Critzer, Mar 03 2012 *)
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CROSSREFS
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KEYWORD
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nonn,tabf,changed
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AUTHOR
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STATUS
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approved
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