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 A108429 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 down steps (d). 1
 1, 0, 1, 1, 0, 0, 2, 5, 3, 0, 0, 0, 5, 21, 28, 12, 0, 0, 0, 0, 14, 84, 180, 165, 55, 0, 0, 0, 0, 0, 42, 330, 990, 1430, 1001, 273, 0, 0, 0, 0, 0, 0, 132, 1287, 5005, 10010, 10920, 6188, 1428, 0, 0, 0, 0, 0, 0, 0, 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Row n contains 2n+1 terms, the first n of which are equal to 0. Row sums yield A027307. T(n,n) = A000108(n) (the Catalan numbers). T(n,2n) = A001764(n) = binomial(3n,n)/(2n+1). Except for the 0's, the same as A104978. Number of d steps in all paths from (0,0) to (3n,0) is given by A108430. LINKS Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370. FORMULA T(n,k) = binomial(n,2n-k)*binomial(n+k, n-1)/n. G.f.: G = G(t, z) satisfies G=1+tzG^2*(1+tG). EXAMPLE Example T(2,3) = 5 because we have udUdd, uUddd, Uddud, Ududd and Uuddd. Triangle begins: 1; 0,1,1; 0,0,2,5,3; 0,0,0,5,21,28,12; ... MAPLE a:=proc(n, k) if n=0 and k=0 then 1 elif n=0 then 0 elif k=0 then 0 else binomial(n, 2*n-k)*binomial(n+k, n-1)/n fi end: for n from 0 to 8 do seq(a(n, k), k=0..2*n) od; # yields sequence in triangular form CROSSREFS Cf. A027307, A000108, A001764, A104978, A108430. Sequence in context: A229029 A055385 A160503 * A130280 A160127 A011035 Adjacent sequences:  A108426 A108427 A108428 * A108430 A108431 A108432 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 03 2005 STATUS approved

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Last modified October 15 13:06 EDT 2019. Contains 328030 sequences. (Running on oeis4.)