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A114608 Triangle read by rows: T(n,k) is the number of bicolored Dyck paths of semilength n and having k peaks of the form ud (0 <= k <= n). A bicolored Dyck path is a Dyck path in which each up-step is of two kinds: u and U. 1
1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 45, 96, 66, 16, 1, 197, 501, 450, 170, 25, 1, 903, 2668, 2955, 1520, 365, 36, 1, 4279, 14407, 18963, 12355, 4165, 693, 49, 1, 20793, 78592, 119812, 94528, 41230, 9856, 1204, 64, 1, 103049, 432073, 748548, 693588, 372078, 117054 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums yield A052701. Column 0 yields the little Schroeder numbers (A001003). Sum_{k=0..n} k*T(n,k) = A069720(n).

Triangle T(n,k), 0 <= k <= n, read by rows; given by [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 23 2005

LINKS

Table of n, a(n) for n=0..50.

Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.

FORMULA

T(n,k) = (1/n)*binomial(n,k)*Sum_{j=0..n-k} 2^j*binomial(n, j+1)*binomial(n-k, j) (k <= n-1); T(n, n)=1.

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

EXAMPLE

T(3,2)=9 because we have (ud)(ud)Ud, (ud)Ud(ud), Ud(ud)(ud), (ud)u(ud)d,

(ud)U(ud)d, u(ud)d(ud), U(ud)d(ud), u(ud)(ud)d and U(ud)(ud)d (the ud peaks are shown between parentheses).

Triangle starts:

   1;

   1,  1;

   3,  4,  1;

  11, 19,  9,  1;

  45, 96, 66, 16,  1;

MAPLE

T:=proc(n, k) if k<=n-1 then (1/n)*binomial(n, k)*sum(2^j*binomial(n, j+1)*binomial(n-k, j), j=0..n-k) elif k=n then 1 else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

MATHEMATICA

T[n_, k_] := If[k <= n-1, (1/n)*Binomial[n, k]*Sum[2^j*Binomial[n, j+1]* Binomial[n-k, j], {j, 0, n-k}], If[k == n, 1, 0]];

Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 11 2018, from Maple *)

CROSSREFS

Cf. A001003, A052701, A069720.

Sequence in context: A138263 A147721 A172094 * A154602 A216154 A325174

Adjacent sequences:  A114605 A114606 A114607 * A114609 A114610 A114611

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 15 2005

STATUS

approved

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Last modified April 20 20:26 EDT 2021. Contains 343137 sequences. (Running on oeis4.)