A108074
Triangle in A071944 with rows reversed.
Original entry on oeis.org
1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 19, 16, 9, 4, 1, 63, 54, 31, 14, 5, 1, 219, 188, 111, 52, 20, 6, 1, 787, 676, 405, 197, 80, 27, 7, 1, 2897, 2492, 1508, 752, 320, 116, 35, 8, 1, 10869, 9361, 5712, 2900, 1276, 489, 161, 44, 9, 1, 41414, 35702, 21933, 11296, 5095, 2034, 714
Offset: 0
Triangle begins:
1;
1, 1;
2, 2, 1;
6, 5, 3, 1;
19, 16, 9, 4, 1; ...
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).
Original entry on oeis.org
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
Offset: 0
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).
- 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
-
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:
-
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 *)
A071946
Triangle T(n,k) 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 = (3,1).
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 6, 13, 19, 19, 1, 8, 23, 44, 63, 63, 1, 10, 37, 87, 156, 219, 219, 1, 12, 55, 155, 330, 568, 787, 787, 1, 14, 77, 255, 629, 1260, 2110, 2897, 2897, 1, 16, 103, 395, 1111, 2527, 4856, 7972, 10869, 10869, 1, 18, 133, 583, 1849, 4706, 10130, 18889, 30545, 41414, 41414
Offset: 0
Triangle T(n,k) begins:
1;
1, 1;
1, 2, 2;
1, 4, 6, 6;
1, 6, 13, 19, 19;
...
-
T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
`if`(k<0 or nAlois P. Heinz, May 05 2023
-
T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
If[k < 0 || n < k, 0, T[n-1, k] + T[n, k-1] + T[n-3, k-1]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 25 2025, after Alois P. Heinz *)
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