OFFSET
0,1
COMMENTS
The first array in the series is Pascal's triangle, A007318. The initial partition for each subsequent array in the series is chosen as described in A053445. When cells are squared, as in A008459, row sums yield 1, 2, 6, 24, ... (A000142). E.g., (1 + 16 + 36 + 16 + 1) + (25 + 25) = 70 + 50 = 120 using row five from A007318 and row two from this array.
Number of lattice paths from (0,0) to (n,k) in exactly n+k+2 steps. Moves allowed: (0,1), (0,-1), (-1,0) and (1,0). Paths must stay in Quadrant I but may touch the axes. - Juan Luis Vargas-Molina, Mar 03 2014
REFERENCES
Stanton and White, Constructive Combinatorics, 1986, pp. 84, 91.
LINKS
Cyann Donnot, Antoine Genitrini, Yassine Herida, Unranking Combinations Lexicographically: an efficient new strategy compared with others, hal-02462764 [cs] / [cs.DS] / [math] / [math.CO], 2020.
Antoine Genitrini and Martin Pépin, Lexicographic unranking of combinations revisited, hal-03040740v2 [cs.DM] [cs.DS] [math.CO], 2020.
Juan Luis Vargas-Molina, C++ Algorithm
FORMULA
T(row, col) = T(row-1, col-1) + T(row-1, col) + A007318(row+2, col+1). - Corrected by R. J. Mathar, Jan 27 2011
T(n,k) = (k+1)(n+k+4)FallingFactorial(n+k+2,n)/((n+1)!+n!) n,k >= 0. - Juan Luis Vargas-Molina, Mar 03 2014
EXAMPLE
a(5) = 16 because we can write T(2,2) = T(1,2) + T(1,1) + A007318(4,2) = 5 + 5 + 6.
2;
5, 5;
9, 16, 9;
14, 35, 35, 14;
20, 64, 90, 64, 20;
27, 105, 189, 189, 105, 27;
T(n,k) as a grid for the first few lattice points. Where T(0,0)=2 since there are two ways to get to (0,0) with "0+0+2" steps under the move restrictions.
k = 0 1 2 3 4 5
T(0,k) = 2 5 9 14 20 27
T(1,k) = 5 16 35 64 105 160
T(2,k) = 9 35 90 189 350 594
T(3,k) = 14 64 189 448 924 1728
T(4,k) = 20 105 350 924 2100 4290
T(5,k) = 27 160 594 1728 4290 9504
- Juan Luis Vargas-Molina, Mar 03 2014
MAPLE
A059797 := proc(n, k) option remember; if n <0 or k<0 or k>n then 0; else procname(n-1, k-1)+procname(n-1, k)+binomial(n+2, k+1) ; end if; end proc:
seq(seq( A059797(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Jan 27 2011
MATHEMATICA
t[n_, k_] /; (n < 0 || k < 0 || k > n) = 0; t[n_, k_] := t[n, k] = t[n - 1, k - 1] + t[n - 1, k] + Binomial[n + 2, k + 1]; Flatten[ Table[ t[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Dec 20 2011, after R. J. Mathar *)
T[n_, k_] := (k + 1)(n + k +4) FactorialPower[k + n + 2, n]/(n! + (n + 1)!)
MatrixForm[Table[T[n, k], {n, 0, 10}, {k, 0, 10}]] (* Juan Luis Vargas-Molina, Mar 03 2014 *)
CROSSREFS
KEYWORD
AUTHOR
Alford Arnold, Feb 22 2001
STATUS
approved