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 A071943 Triangle T(n,k) (n>=0, 0 <= k <= n) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R=(1,0), V=(0,1) and D=(1,2). 6
 1, 1, 1, 1, 2, 3, 1, 3, 7, 9, 1, 4, 12, 24, 31, 1, 5, 18, 46, 89, 113, 1, 6, 25, 76, 183, 342, 431, 1, 7, 33, 115, 323, 741, 1355, 1697, 1, 8, 42, 164, 520, 1376, 3054, 5492, 6847, 1, 9, 52, 224, 786, 2326, 5900, 12768, 22669, 28161, 1, 10, 63, 296, 1134, 3684, 10370 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS For another interpretation of this array see the Example section. LINKS Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened) James East, Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018. D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320. N. J. A. Sloane, Rows 0 through 100 N. J. A. Sloane, Illustration of the initial terms of the U(n,k) array FORMULA G.f.=(1-q)/[z(2t+2t^2z-1+q)], where q=sqrt(1-4tz-4t^2z^2). Define T(0,0)=1 and T(n,k)=0 for k<0 and k >n. Then the array is generated by the recurrence T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k-2). For example, T(5,3) = 46 = T(5,2) + T(4,3) + T(4,1) = 18 + 24 + 4. - N. J. A. Sloane, Mar 28 2013 EXAMPLE T(3,2)=7 because we have RRRVV, RRVRV, RRVVR, RVRRV, RVRVR, RRD and RDR. Array begins: 1, 1, 1, 1, 2, 3, 1, 3, 7, 9, 1, 4, 12, 24, 31, 1, 5, 18, 46, 89, 113, 1, 6, 25, 76, 183, 342, 431, 1, 7, 33, 115, 323, 741, 1355, 1697, ... Equivalently, let U(n,k) (for n >= 0, 0 <= k <= n) be the number of walks from (0,0) to (n,k) using steps (1,1), (1,-1) and (0,-1). Then U(n,n-k) = T(n,k). The U(n,k) array begins: 4:  0  0  0  0  1  5 ... 3:  0  0  0  1  4 18 ... 2:  0  0  1  3 12 46 ... 1:  0  1  2  7 24 89 ... 0:  1  1  3  9 31 113 ... ------------------------- k/n:0  1  2  3  4  5 ... The recurrence for this version is: U(0,0)=1, U(n,k)=0 for k>n or k<0; U(n,k) = U(n,k+1) + U(n-1,k+1) + U(n,k-1). E.g. 46 = 18 + 4 + 24. Also U(n,0) = A052709(n-1). MAPLE U:=proc(n, k) option remember; if (n < 0) then RETURN(0); elif (n=0) then    if (k=0) then RETURN(1); else RETURN(0); fi; elif (k>n or k<0) then RETURN(0); else RETURN(U(n, k+1)+U(n-1, k+1)+U(n-1, k-1)); fi; end; for n from 0 to 20 do lprint( [seq(U(n, n-i), i=0..n)] ); od: MATHEMATICA t[0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, k] + t[n-1, k-2]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2014, after N. J. A. Sloane *) CROSSREFS Diagonal entries yield A052709. Row sums are A071356. Related arrays: A071944, A071945, A071946. Sequence in context: A208338 A236918 A152821 * A062869 A102473 A011117 Adjacent sequences:  A071940 A071941 A071942 * A071944 A071945 A071946 KEYWORD nonn,easy,tabl AUTHOR N. J. A. Sloane, Jun 15 2002 EXTENSIONS Edited by Emeric Deutsch, Dec 21 2003 Edited by N. J. A. Sloane, Mar 28 2013 STATUS approved

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Last modified April 23 05:42 EDT 2021. Contains 343199 sequences. (Running on oeis4.)