A243921 - OEIS

%I #7 Jun 15 2014 22:40:58

%S 1,1,6,53,612,8676,145268,2798355,60852004,1472460760,39202586348,

%T 1138006266618,35750917265544,1207874695612336,43655110115967528,

%U 1680097198812367783,68578132320350944324,2958457556868808457800,134469635178557071054492,6421829932908536633173110

%N G.f.: 1 = Sum_{n>=0} a(n) * x^n*(1-x)^(n+1) / Product_{k=1..n} (1 + 2*(k+1)*x).

%C  Triangle T = A243920 is generated by sums of matrix powers of itself such that:

%C T(n,k) = Sum_{j=1..n-k-1} [T^j](n-1,k) with T(n+1,n) = 2*n+1 and T(n,n)=0 for n>=0.

%F a(n) = A243920(n+1,1) / 3.

%e G.f.: 1 = 1*(1-x) + 1*x*(1-x)^2/(1+2*2*x) + 6*x^2*(1-x)^3/((1+2*2*x)*(1+2*3*x)) + 53*x^3*(1-x)^4/((1+2*2*x)*(1+2*3*x)*(1+2*4*x)) + 612*x^4*(1-x)^5/((1+2*2*x)*(1+2*3*x)*(1+2*4*x)*(1+2*5*x)) +...

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

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A243920, A243922, A243923, A208676.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 15 2014