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A119473 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.) 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Row n has 1+ceiling(n/2) terms.
T(n,0) = Fibonacci(n+1) = A000045(n+1).
T(n,1) = A029907(n).
Sum_{k>=0} k*T(n,k) = A059570(n).
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
REFERENCES
I. Goulden and D. Jackson, Combinatorial Enumeration, John Wiley and Sons, 1983, page 54.
LINKS
R. Grimaldi and S. Heubach, Binary strings without odd runs of zeros, Ars Combinatoria 75 (2005), 241-255.
FORMULA
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).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Dec 07 2011
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
EXAMPLE
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;
From Philippe Deléham, Dec 07 2011: (Start)
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)
MAPLE
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):
seq(T(n), n=0..14); # Alois P. Heinz, May 23 2013
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A087824 A369475 A008951 * A336889 A002122 A105689
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, May 22 2006
STATUS
approved

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Last modified March 19 07:49 EDT 2024. Contains 370958 sequences. (Running on oeis4.)