%I #11 Mar 24 2018 16:53:28
%S 1,2,5,16,60,254,1188,6043,33080,193249,1197001,7819995,53648847,
%T 385090323,2883045424,22451716833,181437812058,1518374146260,
%U 13133970646948,117235109969112,1078235776311405,10204120439288725,99244762587719585,990878067150790140,10145281310155565842,106420501631411705747,1142671059786354295966,12548652816798990883431,140839029768184796119004
%N G.f.: Sum_{n>=0} (1 + (1+x)^n)^n * x^n.
%H Paul D. Hanna, <a href="/A301306/b301306.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: Sum_{n>=0} x^n * (1+x)^(n^2) / (1 - x*(1+x)^n)^(n+1).
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1+x)^(n*k).
%F a(n) = Sum_{j=0..n} Sum_{k=0..n-j} binomial(n-j, k) * binomial((n-j)*k, j).
%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 16*x^3 + 60*x^4 + 254*x^5 + 1188*x^6 + 6043*x^7 + 33080*x^8 + 193249*x^9 + 1197001*x^10 + ...
%e such that
%e A(x) = 1 + (1 + (1+x))*x + (1 + (1+x)^2)^2*x^2 + (1 + (1+x)^3)^3*x^3 + (1 + (1+x)^4)^4*x^4 + (1 + (1+x)^5)^5*x^5 + (1 + (1+x)^6)^6*x^6 + ...
%o (PARI) {a(n) = my(A=1); A = sum(k=0,n, (1 + (1+x)^k +x*O(x^n))^k * x^k ); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = sum(j=0,n, sum(k=0,n-j, binomial(n-j,k) * binomial((n-j)*k,j) ))}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A301465.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 21 2018