A061187 - OEIS

%I #4 Mar 31 2012 13:20:05

%S 3,-2,6,2,-4,9,39,-57,30,-8,12,136,-96,-84,104,-32,15,320,293,-1260,

%T 1155,-530,160,-32,18,618,2118,-4242,890,2718,-2652,1088,-192,21,1057,

%U 7224,-5037,-19208,33383,-23793,9534,-2632,672

%N Staircase of coefficients of polynomials used for column g.f.s of triangle A060924.

%C a(n,m) is coefficient of x^m of polynomial pLo(n+1,x) := (((1+x)+(3-2*x)*sqrt(x))^(n+1) - ((1+x)-(3-2*x)*sqrt(x))^(n+1))/(2*sqrt(x)) of degree n+1+floor(n/2)= A001651(n). pLo(n+1,x)= sum(binomial(n+1,2*j+1)*(1+x)^(n-2*j)*(3-2*x)^(2*j+1)*x^j,j=0..floor(n/2)), n >= 0.

%C pLo(m+1,x) appears as numerator polynomial of the g.f. for column m >= 0 of the triangle A060924 (even part of bisection of Lucas triangle).

%F a(n, m)= sum(3*(-9/2)^j*binomial(n+1, 2*j+1)*sum((-3/2)^(k-m)*binomial(n-2*j, k) *binomial(2*j+1, m-k-j), k=max(0, m-3*j-1)..n-2*j), j=0..floor(n/2)), 0<= m <= n+1+floor(n/2); else 0.

%e {3, -2}; {6, 2, -4}; {9, 39, -57, 30, -8}; ...; pLo(2, x)= 6+2*x-4*x^2= 2*(1+x)*(3-2*x).

%Y A061186 (companion staircase).

%K sign,easy,tabf

%O 0,1

%A _Wolfdieter Lang_, Apr 20 2001