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A111808 Left half of trinomial triangle (A027907), triangle read by rows.

%I A111808 #27 Feb 16 2025 08:32:58
%S A111808 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,
%T A111808 141,1,7,28,77,161,266,357,393,1,8,36,112,266,504,784,1016,1107,1,9,
%U A111808 45,156,414,882,1554,2304,2907,3139,1,10,55,210,615,1452,2850,4740,6765,8350
%N A111808 Left half of trinomial triangle (A027907), triangle read by rows.
%C A111808 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.
%C A111808 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
%D A111808 Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.
%H A111808 G. C. Greubel, <a href="/A111808/b111808.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A111808 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialTriangle.html">Trinomial Triangle</a>
%H A111808 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%F A111808 (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);
%F A111808 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.
%F A111808 T(n, k) = GegenbauerC(k, -n, -1/2). - _Peter Luschny_, May 09 2016
%p A111808 T := (n,k) -> simplify(GegenbauerC(k, -n, -1/2)):
%p A111808 for n from 0 to 9 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, May 09 2016
%t A111808 Table[GegenbauerC[k, -n, -1/2], {n,0,10}, {k,0,n}] // Flatten (* _G. C. Greubel_, Feb 28 2017 *)
%Y A111808 Row sums give A027914; central terms give A027908;
%Y A111808 T(n, 0) = 0;
%Y A111808 T(n, 1) = n for n>1;
%Y A111808 T(n, 2) = A000217(n) for n>1;
%Y A111808 T(n, 3) = A005581(n) for n>2;
%Y A111808 T(n, 4) = A005712(n) for n>3;
%Y A111808 T(n, 5) = A000574(n) for n>4;
%Y A111808 T(n, 6) = A005714(n) for n>5;
%Y A111808 T(n, 7) = A005715(n) for n>6;
%Y A111808 T(n, 8) = A005716(n) for n>7;
%Y A111808 T(n, 9) = A064054(n-5) for n>8;
%Y A111808 T(n, n-5) = A098470(n) for n>4;
%Y A111808 T(n, n-4) = A014533(n-3) for n>3;
%Y A111808 T(n, n-3) = A014532(n-2) for n>2;
%Y A111808 T(n, n-2) = A014531(n-1) for n>1;
%Y A111808 T(n, n-1) = A005717(n) for n>0;
%Y A111808 T(n, n) = central terms of A027907 = A002426(n).
%K A111808 nonn,tabl
%O A111808 1,5
%A A111808 _Reinhard Zumkeller_, Aug 17 2005
%E A111808 Corrected and edited by _Johannes W. Meijer_, Oct 05 2010