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%I #9 Feb 11 2015 06:34:26
%S 1,-4,1,15,-6,1,-56,28,-8,1,210,-120,45,-10,1,-792,495,-220,66,-12,1,
%T 3003,-2002,1001,-364,91,-14,1,-11440,8008,-4368,1820,-560,120,-16,1,
%U 43758,-31824,18564,-8568,3060,-816,153,-18,1,-167960,125970,-77520
%N Triangle read by rows: alternating binomial coefficients with signs.
%C Alternating binomial coefficients in the closed form expression for sequence A156289.
%C The Example lines below show the connection with Pascal's triangle A007318.
%D T. Myers and L. Shapiro, Some applications of the sequence 1, 5, 22, 93, 386, ... to Dyck paths and ordered trees, Congressus Numerant., 204 (2010), 93-104.
%F R(k,j)=(-1)^(k+j)*Binomial(2k,k+j), for 1<= j<=k, and 0 otherwise.
%e R(2,1)=-4, R(3,3)=1, R(4,2)=28.
%e Here is Pascal's triangle with the entries in the present triangle preceded by a *:
%e ......................1
%e .....................1, 1
%e ...................1, 2,*1
%e .................1, 3, 3, 1
%e ................1, 4, 6,*4,*1
%e ..............1, 5, 10, 10, 5, 1
%e ............1, 6, 15, 20,*15,*6,*1
%e ..........1, 7, 21, 35, 35, 21, 7, 1
%e ........1, 8, 28, 56, 70,*56,*28,*8,*1
%e ...
%t R[m_] := Flatten[Table[(-1)^(k + j) Binomial[2 k, k + j], {k, 1, m}, {j, 1, k}]]
%Y Coefficient factor in elements of sequence A156289, the inverse of lower triangular matrix A156308.
%Y Cf. A007318.
%K easy,sign,tabl
%O 1,2
%A _Hartmut F. W. Hoft_, Feb 07 2009
%E Edited by _N. J. A. Sloane_, Apr 05 2011
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