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A059797
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Second in a series of arrays counting standard tableaux by partition type.
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9
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2, 5, 5, 9, 16, 9, 14, 35, 35, 14, 20, 64, 90, 64, 20, 27, 105, 189, 189, 105, 27, 35, 160, 350, 448, 350, 160, 35, 44, 231, 594, 924, 924, 594, 231, 44, 54, 320, 945, 1728, 2100, 1728, 945, 320, 54, 65, 429, 1430, 3003, 4290, 4290, 3003, 1430, 429, 65
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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Stanton and White, Constructive Combinatorics, 1986, pp. 84, 91.
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MAPLE
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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:
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MATHEMATICA
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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)!)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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