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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

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

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

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

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Last modified July 6 17:32 EDT 2020. Contains 335479 sequences. (Running on oeis4.)