The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A109983 Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps. 6
 1, 0, 1, 2, 0, 0, 1, 6, 6, 0, 0, 0, 1, 12, 30, 20, 0, 0, 0, 0, 1, 20, 90, 140, 70, 0, 0, 0, 0, 0, 1, 30, 210, 560, 630, 252, 0, 0, 0, 0, 0, 0, 1, 42, 420, 1680, 3150, 2772, 924, 0, 0, 0, 0, 0, 0, 0, 1, 56, 756, 4200, 11550, 16632, 12012, 3432 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A Delannoy path of length n is a path from (0, 0) to (n, n), consisting of steps E = (1,0), N = (0,1) and D = (1,1)). Row n has 2*n+1 terms, the first n of which are 0. Row sums are the central Delannoy numbers (A001850). Column sums are the central trinomial coefficients (A002426). LINKS Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5. FORMULA T(n, k) = binomial(n, 2*n-k) binomial(k, n). T(n, k) = A104684(n, 2*n-k). G.f.: 1/sqrt((1 - t*z)^2 - 4*z*t^2). T(n, 2*n) = binomial(2*n, n) (A000984). Sum_{k=0..n} k*T(n, k) = A109984(n). T(n, k) = A063007(n, k-n). - Michael Somos, Sep 22 2013 EXAMPLE T(2, 3) = 6 because we have DNE, DEN, NED, END, NDE and EDN. Triangle begins   1;   0,1,2;   0,0,1,6,6;   0,0,0,1,12,30,20;   ... MAPLE T := (n, k)->binomial(n, 2*n-k)*binomial(k, n): for n from 0 to 8 do seq(T(n, k), k=0..2*n) od; # yields sequence in triangular form # Alternative: gf := ((1 - x*y)^2 - 4*x^2*y)^(-1/2): yser := series(gf, y, 12): ycoeff := n -> coeff(yser, y, n): row := n -> seq(coeff(expand(ycoeff(n)), x, k), k=0..2*n): seq(row(n), n=0..7); # Peter Luschny, Oct 28 2020 PROG (PARI) {T(n, k) = binomial(n, k-n) * binomial(k, n)} /* Michael Somos, Sep 22 2013 */ (Haskell) a109983 n k = a109983_tabf !! n !! k a109983_row n = a109983_tabf !! n a109983_tabf = zipWith (++) (map (flip take (repeat 0)) [0..]) a063007_tabl -- Reinhard Zumkeller, Nov 18 2014 CROSSREFS Cf. A001850, A002426, A000984, A063007, A104684, A109984. Sequence in context: A113129 A127826 A228866 * A332409 A289084 A193033 Adjacent sequences:  A109980 A109981 A109982 * A109984 A109985 A109986 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jul 07 2005 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 18 02:38 EDT 2021. Contains 343072 sequences. (Running on oeis4.)