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A111808
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Left half of trinomial triangle (A027907), triangle read by rows.
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30
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1, 1, 1, 1, 2, 3, 1, 3, 6, 7, 1, 4, 10, 16, 19, 1, 5, 15, 30, 45, 51, 1, 6, 21, 50, 90, 126, 141, 1, 7, 28, 77, 161, 266, 357, 393, 1, 8, 36, 112, 266, 504, 784, 1016, 1107, 1, 9, 45, 156, 414, 882, 1554, 2304, 2907, 3139, 1, 10, 55, 210, 615, 1452, 2850, 4740, 6765, 8350
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OFFSET
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1,5
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COMMENTS
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Consider a doubly infinite chessboard with squares labeled (n,k), ranks or rows n in Z, files or columns k in Z (Z denotes ...,-2,-1,0,1,2,... ); number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,n-k). - Harrie Grondijs, May 27 2005. Cf. A026300, A114929, A114972.
Triangle of numbers C^(2)(n-1,k), n>=1, of combinations with repetitions from elements {1,2,...,n} over k, such that every element i, i=1,...,n, appears in a k-combination either 0 or 1 or 2 times (cf. also A213742-A213745). - Vladimir Shevelev and Peter J. C. Moses, Jun 19 2012
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REFERENCES
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Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.
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LINKS
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FORMULA
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(1 + x + x^2)^n = Sum(T(n,k)*x^k: 0<=k<=n) + Sum(T(n,k)*x^(2*n-k): 0<=k<n);
T(n, k) = A027907(n, k) = Sum_{i=0,..,(k/2)} binomial(n, n-k+2*i) * binomial(n-k+2*i, i), 0<=k<=n.
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MAPLE
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T := (n, k) -> simplify(GegenbauerC(k, -n, -1/2)):
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, May 09 2016
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MATHEMATICA
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Table[GegenbauerC[k, -n, -1/2], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)
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CROSSREFS
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T(n, 0) = 0;
T(n, 1) = n for n>1;
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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