A111808 - OEIS

%I #26 Feb 28 2017 22:34:32

%S 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 141,1,7,28,77,161,266,357,393,1,8,36,112,266,504,784,1016,1107,1,9,

%U 45,156,414,882,1554,2304,2907,3139,1,10,55,210,615,1452,2850,4740,6765,8350

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

%C 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 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 Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.

%H G. C. Greubel, <a href="/A111808/b111808.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialTriangle.html">Trinomial Triangle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>

%F (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 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 T(n, k) = GegenbauerC(k, -n, -1/2). - _Peter Luschny_, May 09 2016

%p T := (n,k) -> simplify(GegenbauerC(k, -n, -1/2)):

%p for n from 0 to 9 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, May 09 2016

%t Table[GegenbauerC[k, -n, -1/2], {n,0,10}, {k,0,n}] // Flatten (* _G. C. Greubel_, Feb 28 2017 *)

%Y Row sums give A027914; central terms give A027908;

%Y T(n, 0) = 0;

%Y T(n, 1) = n for n>1;

%Y T(n, 2) = A000217(n) for n>1;

%Y T(n, 3) = A005581(n) for n>2;

%Y T(n, 4) = A005712(n) for n>3;

%Y T(n, 5) = A000574(n) for n>4;

%Y T(n, 6) = A005714(n) for n>5;

%Y T(n, 7) = A005715(n) for n>6;

%Y T(n, 8) = A005716(n) for n>7;

%Y T(n, 9) = A064054(n-5) for n>8;

%Y T(n, n-5) = A098470(n) for n>4;

%Y T(n, n-4) = A014533(n-3) for n>3;

%Y T(n, n-3) = A014532(n-2) for n>2;

%Y T(n, n-2) = A014531(n-1) for n>1;

%Y T(n, n-1) = A005717(n) for n>0;

%Y T(n, n) = central terms of A027907 = A002426(n).

%K nonn,tabl

%O 1,5

%A _Reinhard Zumkeller_, Aug 17 2005

%E Corrected and edited by _Johannes W. Meijer_, Oct 05 2010