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A108446 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k peaks of the form ud. 2
1, 1, 1, 4, 5, 1, 20, 32, 13, 1, 113, 223, 135, 26, 1, 688, 1620, 1300, 412, 45, 1, 4404, 12064, 12050, 5350, 1030, 71, 1, 29219, 91335, 109134, 62450, 17575, 2247, 105, 1, 199140, 699689, 973077, 682234, 254625, 49210, 4438, 148, 1, 1385904, 5407744 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums yield A027307. Column 0 yields A108447. T(n,n-1) = A008778(n-1) = n(n^2+6n-1)/6. Number of ud peaks in all paths from (0,0) to (3n,0) is given by A108448.

LINKS

Table of n, a(n) for n=0..46.

Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.

FORMULA

T(n,k) = (1/n) binomial(n, k)*sum(binomial(n-k,j)*binomial(n+2j,k+j-1), j=0..n-k).

G.f.: G = G(t,z) satisfies G = 1+z(G-1+t)G+zG^3.

EXAMPLE

T(2,1) = 5 because we have udUdd, uudd, Uddud, Ududd and Uuddd.

Triangle begins:

1;

1,1;

4,5,1;

20,32,13,1;

113,223,135,26,1;

MAPLE

T:=proc(n, k) if n=0 and k=0 then 1 elif n=0 then 0 elif k=n then 1 elif k=n then 1 else (1/n)*binomial(n, k)*sum(binomial(n-k, j)*binomial(n+2*j, k+j-1), j=0..n-k) fi end: for n from 0 to 9 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

MATHEMATICA

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

Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 19 2018 *)

CROSSREFS

Cf. A027307, A008778, A108447, A108448, A108425, A108426.

Sequence in context: A266699 A234937 A210590 * A283263 A109962 A102230

Adjacent sequences:  A108443 A108444 A108445 * A108447 A108448 A108449

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jun 10 2005

STATUS

approved

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Last modified May 6 09:45 EDT 2021. Contains 343580 sequences. (Running on oeis4.)