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Search: id:A033877
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| A033877 |
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Triangular array associated with Schroeder numbers: T(1,* ) = 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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+0
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; listen)
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OFFSET
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1,3
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COMMENT
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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.
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LINKS
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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.
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FORMULA
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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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MATHEMATICA
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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
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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).
Sequence in context: A063872 A033884 A062344 this_sequence A059369 A098473 A121757
Adjacent sequences: A033874 A033875 A033876 this_sequence A033878 A033879 A033880
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KEYWORD
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nonn,tabl,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net)
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