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 A111808 Left half of trinomial triangle (A027907), triangle read by rows. 30
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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 REFERENCES Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle. LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened Eric Weisstein's World of Mathematics, Trinomial Triangle Eric Weisstein's World of Mathematics, Trinomial Coefficient FORMULA (1 + x + x^2)^n = Sum(T(n,k)*x^k: 0<=k<=n) + Sum(T(n,k)*x^(2*n-k): 0<=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 MATHEMATICA Table[GegenbauerC[k, -n, -1/2], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Feb 28 2017 *) CROSSREFS Row sums give A027914; central terms give A027908; T(n, 0) = 0; T(n, 1) = n for n>1; T(n, 2) = A000217(n) for n>1; T(n, 3) = A005581(n) for n>2; T(n, 4) = A005712(n) for n>3; T(n, 5) = A000574(n) for n>4; T(n, 6) = A005714(n) for n>5; T(n, 7) = A005715(n) for n>6; T(n, 8) = A005716(n) for n>7; T(n, 9) = A064054(n-5) for n>8; T(n, n-5) = A098470(n) for n>4; T(n, n-4) = A014533(n-3) for n>3; T(n, n-3) = A014532(n-2) for n>2; T(n, n-2) = A014531(n-1) for n>1; T(n, n-1) = A005717(n) for n>0; T(n, n) = central terms of A027907 = A002426(n). Sequence in context: A209569 A176850 A208516 * A247046 A081422 A213742 Adjacent sequences:  A111805 A111806 A111807 * A111809 A111810 A111811 KEYWORD nonn,tabl AUTHOR Reinhard Zumkeller, Aug 17 2005 EXTENSIONS Corrected and edited by Johannes W. Meijer, Oct 05 2010 STATUS approved

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Last modified March 26 22:42 EDT 2019. Contains 321565 sequences. (Running on oeis4.)