A177350 - OEIS

%I #11 Mar 12 2014 16:37:17

%S -1,0,2,1,-1,3,2,0,4,5,4,1,-1,5,8,7,3,0,6,12,13,12,7,1,-1,7,17,21,20,

%T 14,4,0,8,23,33,34,33,26,11,1,-1,9,30,50,55,54,46,25,5,0,10,38,73,88,

%U 89,88,79,51,16,1,-1

%N Triangle read by rows: T(n,k)=Sum(binomial(n - (m + k), m + k), (m, 1, floor[n/2 + 1]))

%C Row sums are:{-1, 0, 2, 5, 13, 23, 50, 83, 168, 274, 532,...}.

%C An effort to generalize the Lucas Fibonacci sum: m limit seems to be off. [Connection with Lucas or Fibonacci sequences is not clear to me, and what does "off" mean? - _N. J. A. Sloane_, Dec 16 2010]

%C For comparison: the incomplete generalized Fibonacci Polynomials [arXiv:1308.4192] F_n(k) = sum_{j=0..k} binomial(n-1-j,j) start with a leading column of 1's as:

%C 1;

%C 1;

%C 1 2;

%C 1 3;

%C 1 4 5;

%C 1 5 8;

%C 1 6 12 13;

%C 1 7 17 21;

%C 1 8 23 33 34;

%C 1 9 30 50 55; - _R. J. Mathar_, Aug 23 2013

%F k between -Floor[n/2]and Floor[n/2]:

%F t(n,k)=Sum(Binomial(n - (m + k), m + k), {m, 1, Floor[n/2 + 1]))

%e -1;

%e 0;

%e 2, 1, -1;

%e 3, 2, 0;

%e 4, 5, 4, 1, -1;

%e 5, 8, 7, 3, 0;

%e 6, 12, 13, 12, 7, 1, -1;

%e 7, 17, 21, 20, 14, 4, 0;

%e 8, 23, 33, 34, 33, 26, 11, 1, -1;

%e 9, 30, 50, 55, 54, 46, 25, 5, 0;

%e 10, 38, 73, 88, 89, 88, 79, 51, 16, 1, -1;

%t w[n_, m_, k_] = Binomial[n - (m + k), m + k];

%t t[n_, k_] := Sum[w[n, m, k], {m, 1, Floor[n/2 + 1]}];

%t Table[Table[t[n, k], {k, -Floor[n/2], Floor[n/2]}], {n, 0, 10}];

%t Flatten[%]

%Y Cf. A000045

%K sign,tabf

%O 0,3

%A _Roger L. Bagula_, Dec 10 2010