A102917 - OEIS

%I #3 Mar 30 2012 18:36:44

%S 1,1,1,3,7,36,139,1036,5711,56355,408354,5045370,45605881,679409158,

%T 7390305396,129195427716,1647470410551,33114233390505,485292763088275,

%U 11038606786054201,183049273155939442,4652371578279864792

%N Column 0 of triangle A102916.

%C Also equals the interleaving of A082162 with A102921, which equal column 0 of triangle A102098 and its matrix square (A102920), respectively.

%F G.f.: 1 = Sum_{n>=0}(a(2*n)+a(2*n+1)*x)*x^(2*n)*Product_{k=1..n+1}(1-k*x) where a(2*n)=A082162(n) and a(2*n+1)=A102921(n).

%e 1 = 1*(1-x) + 1*x*(1-x) + 1*x^2*(1-x)(1-2x) + 3*x^3*(1-x)(1-2x)

%e + 7*x^4*(1-x)(1-2x)(1-3x) + 36*x^5*(1-x)(1-2x)(1-3x)

%e + 139*x^6*(1-x)(1-2x)(1-3x)(1-4x) + 1036*x^7*(1-x)(1-2x)(1-3x)(1-4x) + ...

%e + A082162(n)*x^(2n)*(1-x)(1-2x)*..*(1-(n+1)x)

%e + A102921(n)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+1)x) + ...

%o (PARI) {a(n)=if(n==0,1,polcoeff(1-sum(k=0,n-1,a(k)*x^k*prod(j=1,k\2+1,1-j*x+x*O(x^n))),n))}

%Y Cf. A102916, A082162, A102921, A102098, A102920.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jan 21 2005