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A191318 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) having pyramid weight equal to k. 1
1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 10, 4, 1, 6, 16, 12, 1, 7, 24, 30, 8, 1, 8, 33, 56, 28, 1, 9, 44, 98, 84, 16, 1, 10, 56, 152, 179, 64, 1, 11, 70, 228, 358, 224, 32, 1, 12, 85, 320, 618, 536, 144, 1, 13, 102, 440, 1030, 1206, 576, 64, 1, 14, 120, 580, 1580, 2292, 1528, 320, 1, 15, 140, 754, 2370, 4202, 3820, 1440, 128 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

A pyramid in a dispersed Dyck path is a factor of the form U^h D^h, h being the height of the pyramid and U=(1,1), D=(1,-1). A pyramid in a dispersed Dyck path w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a dispersed Dyck path is the sum of the heights of its maximal pyramids.

Row n has 1 + floor(n/2) entries.

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

LINKS

Table of n, a(n) for n=0..80.

A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.

FORMULA

T(n,0) = 1;

T(n,1) = n-1 (n>=1).

T(n,2) = A001859(n-3) (n>=4).

Sum_{k>=0} k*T(n,k) = A191319(n).

G.f.: G=G(t,z) satisfies z*(1-z)*(z-1+2*t*z^2)*G^2 + (1-z)*(z-1+2*t*z^2)*G+1-t*z^2=0.

EXAMPLE

T(6,2)=10 because we have HH(UD)(UD), HH(UUDD), H(UD)H(UD), H(UD)(UD)H, H(UUDD)H, (UD)HH(UD), (UD)H(UD)H, (UD)(UD)HH, (UUDD)HH, and U(UD)(UD)D, where U=(1,1), D=(1,-1), H=(1,0); the maximal pyramids are shown between parentheses.

Triangle starts:

  1;

  1;

  1,  1;

  1,  2;

  1,  3,  2;

  1,  4,  5;

  1,  5, 10,  4;

  1,  6, 16, 12;

  1,  7, 24, 30,  8;

MAPLE

a := (z-1)*(2*t*z^2+z-1): c := -1+t*z^2: eq := a*z*G^2+a*G+c: f := RootOf(eq, G): fser := simplify(series(f, z = 0, 20)): for n from 0 to 16 do P[n] := sort(expand(coeff(fser, z, n))) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

CROSSREFS

Cf. A001405, A001859, A191319.

Sequence in context: A049280 A108786 A008315 * A293600 A191395 A183917

Adjacent sequences:  A191315 A191316 A191317 * A191319 A191320 A191321

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jun 01 2011

STATUS

approved

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Last modified December 14 09:41 EST 2019. Contains 329979 sequences. (Running on oeis4.)