A191393 -id:A191393

Search: a191393 -id:a191393

Displaying 1-2 of 2 results found.

page 1

A191392

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 k base pyramids. A base pyramid is a factor of the form U^j D^j (j > 0), starting at the horizontal axis. Here U = (1,1) and D = (1,-1).

+10
3

1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 6, 3, 2, 9, 8, 1, 3, 12, 16, 4, 8, 18, 30, 13, 1, 13, 26, 50, 32, 5, 31, 47, 83, 71, 19, 1, 49, 80, 132, 140, 55, 6, 109, 162, 223, 263, 140, 26, 1, 170, 292, 377, 468, 316, 86, 7, 371, 592, 693, 830, 665, 246, 34, 1, 581, 1064, 1264, 1456, 1314, 622, 126, 8

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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

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

T(n,0) = A191393(n).

Sum_{k>=0} k*T(n,k) = A191394(n).

LINKS

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened

FORMULA

G.f.: G(t,z) = (2*(1 - z^2))/(1 - 2*z + z^2 + 2*z^3 - 2*t*z^2 + (1 - z^2)*sqrt(1 - 4*z^2)).

EXAMPLE

T(5,2)=3 because we have H(UD)(UD), (UD)H(UD), and (UD)(UD)H, where U=(1,1), D=(1,-1), H=(1,0) (the base pyramids are shown between parentheses).

Triangle starts:

1, 1;

1, 2;

1, 4, 1;

1, 6, 3;

2, 9, 8, 1;

3, 12, 16, 4;

8, 18, 30, 13, 1;

MAPLE

G := (2*(1-z^2))/(1-2*z+z^2+2*z^3-2*t*z^2+(1-z^2)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

MATHEMATICA

CoefficientList[CoefficientList[Series[(2*(1 - z^2))/(1 - 2*z + z^2 + 2*z^3 - 2*t*z^2 + (1 - z^2)*Sqrt[1 - 4*z^2]), {z, 0, 10}, {t, 0, 10}], z], t] // Flatten (* G. C. Greubel, Mar 29 2017 *)

CROSSREFS

Cf. A001405, A191393, A191394.

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jun 04 2011

STATUS

approved

A191395

Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., of Motzkin paths of length n with no (1,0)-steps at positive heights) for which the sum of the heights of its base pyramid is k. A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis. Here U=(1,1) and D=(1,-1).

+10
1

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 2, 5, 9, 4, 3, 6, 14, 12, 8, 9, 20, 25, 8, 13, 14, 27, 44, 28, 31, 29, 40, 70, 66, 16, 49, 54, 62, 104, 129, 64, 109, 115, 116, 159, 225, 168, 32, 170, 212, 217, 250, 363, 360, 144, 371, 430, 445, 444, 581, 681, 416, 64, 581, 772, 854, 820, 938, 1182, 968, 320

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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

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

T(n,0) = A191393(n).

Sum_{k>=0} k*T(n,k) = A191397(n).

LINKS

Table of n, a(n) for n=0..71.

FORMULA

G.f.: G=G(t,z) satisfies G = 1+z*G+z^2*G*(c+t/(1-t*z^2)-1/(1-z^2)), where c = (1-sqrt(1-4*z^2))/(2*z^2) (the Catalan function with argument z^2).

EXAMPLE

T(4,2)=2 because we have UDUD and UUDD, where U=(1,1), D=(1,-1), H=(1,0).

Triangle starts:

1, 1;

1, 2;

1, 3, 2;

1, 4, 5;

2, 5, 9, 4;

3, 6, 14, 12;

8, 9, 20, 25, 8;