A191530 - OEIS

A191530

Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of semilength n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) for which the sum of the lengths of the initial and final horizontal segments is k (0<=k<=n).

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 1, 4, 0, 4, 0, 1, 6, 2, 6, 0, 5, 0, 1, 5, 12, 3, 8, 0, 6, 0, 1, 20, 10, 18, 4, 10, 0, 7, 0, 1, 21, 40, 15, 24, 5, 12, 0, 8, 0, 1, 70, 42, 60, 20, 30, 6, 14, 0, 9, 0, 1, 84, 140, 63, 80, 25, 36, 7, 16, 0, 10, 0, 1, 252, 168, 210, 84, 100, 30, 42, 8, 18, 0, 11, 0, 1

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OFFSET

0,8

COMMENTS

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

T(n,0) = A191529(n).

Sum_{k>=0} k*T(n,k) = A191531(n).

LINKS

Table of n, a(n) for n=0..90.

FORMULA

T(2n,0) = binomial(2n-2,n-1) (n>=1); T(2n+1,0) = binomial(2n-1,n-2) (n>=1).

T(n,k) = (k+1)T(n-k,0).

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

EXAMPLE

T(5,3)=4 because we have HHHUD, HHUDH, HUDHH, and UDHHH, where U=(1,1), H=(1,0), and D=(1,-1).

Triangle starts:

0, 1;

1, 0, 1;

0, 2, 0, 1;

2, 0, 3, 0, 1;

1, 4, 0, 4, 0, 1;

6, 2, 6, 0, 5, 0, 1;

MAPLE

q := sqrt(1-4*z^2): G := (2-3*z-t*z+2*t*z^2+(1-t)*z*q)/((1-2*z+q)*(1-t*z)^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

T := proc (n, k) if n < k then 0 elif k = n then 1 elif k = 0 and n = 1 then 0 elif k = 0 and `mod`(n, 2) = 0 then binomial(n-2, (1/2)*n-1) elif k = 0 and `mod`(n, 2) = 1 then binomial(n-2, (1/2)*n-5/2) else (1+k)*T(n-k, 0) end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

MATHEMATICA

T[n_, k_] := Which[n < k, 0,

k == n, 1,

k == 0 && n == 1, 0,

k == 0 && Mod[n, 2] == 0, Binomial[n - 2, n/2 - 1],

k == 0 && Mod[n, 2] == 1, Binomial[n - 2, (n - 5)/2],

True, (1 + k)*T[n - k, 0]];

Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 30 2024, after 2nd Maple program *)

CROSSREFS

Cf. A001405, A191529, A191531.

Sequence in context: A053121 A113408 A242653 * A321435 A173863 A022337

Adjacent sequences: A191527 A191528 A191529 * A191531 A191532 A191533

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jun 07 2011

STATUS

approved