login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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<=ceil(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.

R. Grimaldi and S. Heubach, Binary strings without odd runs of zeros, Ars Combinatoria 75 (2005), 241-255.

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

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<=k<=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;

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... - Philippe Deléham, Dec 07 2011

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

Cf. A000045, A029907, A059570.

Sequence in context: A104567 A087824 A008951 * A002122 A105689 A187200

Adjacent sequences:  A119470 A119471 A119472 * A119474 A119475 A119476

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, May 22 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.