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A033877
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Triangular array associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k<n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).
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13
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1, 1, 2, 1, 4, 6, 1, 6, 16, 22, 1, 8, 30, 68, 90, 1, 10, 48, 146, 304, 394, 1, 12, 70, 264, 714, 1412, 1806, 1, 14, 96, 430, 1408, 3534, 6752, 8558, 1, 16, 126, 652, 2490, 7432, 17718, 33028, 41586, 1, 18, 160, 938, 4080, 14002, 39152, 89898, 164512, 206098, 1, 20, 198
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| The diagonals of this triangle are self-convolutions of the main diagonal A006318 : 1, 2, 6, 22, 90, 394, 1806, . . . - Philippe DELEHAM, May 15 2005
A106579 is in some ways a better version of this sequence, but since this was entered first it will be the main entry for this triangle.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010: (Start)
The triangle sums, see A180662 for their definitions, link Schroeder's triangle, its mirror image is A080247, with (until now just) five sequences, see the crossrefs.
(End)
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LINKS
| T. D. Noe, Rows n=1..50 of triangle, flattened
H. Bottomley, Illustration of initial terms
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), Article 03.1.5, 19 pp.
Kevin Brown, Hipparchus on Compound Statements, 1994-2010. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010]
Johannes. W. Meijer, Famous numbers on a chessboard, 2010. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010]
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FORMULA
| As an upper right triangle: a(n, k) = a(n, k-1)+a(n-1, k-1)+a(n-1, k) if k >= n >= 0 and a(n, k)=0 otherwise.
G.f.: Sum T(n, k)*x^n*y^k = (1-x*y-(x^2*y^2-6*x*y+1)^(1/2)) / (x*(2*y+x*y-1+(x^2*y^2-6*x*y+1)^(1/2))). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 16 2003
Another version of A000007 DELTA [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] = 1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, ..., where DELTA is Deleham's operator defined in A084938.
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MAPLE
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010: (Start)
nmax:=10: kmax:=nmax: for n from 1 to nmax do for k from 1 to kmax do T(n, k):=0 od: od: for k from 1 to kmax do T(1, k):=1 od: for k from 2 to kmax do for n from 2 to k do T(n, k) := T(n, k-1) + T(n-1, k-1) + T(n-1, k) od: od: Tx:=0: for k from 1 to kmax do for n from 1 to k do Tx:=Tx+1: a(Tx):=T(n, k): od: od: seq(a(n), n=1..Tx);
(End)
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MATHEMATICA
| T[ 1, _ ] := 1; T[ n_, k_ ]/; (k<n) := 0; T[ n_, k_ ] := T[ n, k ]=T[ n, k-1 ]+T[ n-1, k-1 ]+T[ n-1, k ];
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CROSSREFS
| Essentially same triangle as A080245 but with rows read in reversed order. Also essentially the same triangle as A106579.
Cf. A008288, A006318, A006319, A006320, A006321, A001003 (row sums), A000007, A084938.
Cf. A026003 (antidiagonal sums).
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010: (Start)
Triangle sums (see the comments): A001003 (Row1); A026003 (Kn11); A006603 (Kn21); A001850 (Kn4); A010683 (Fi2).
(End)
Sequence in context: A033884 A199704 A062344 * A059369 A199530 A098473
Adjacent sequences: A033874 A033875 A033876 * A033878 A033879 A033880
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KEYWORD
| nonn,tabl,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net)
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