A301306 G.f.: Sum_{n>=0} (1 + (1+x)^n)^n * x^n.

1, 2, 5, 16, 60, 254, 1188, 6043, 33080, 193249, 1197001, 7819995, 53648847, 385090323, 2883045424, 22451716833, 181437812058, 1518374146260, 13133970646948, 117235109969112, 1078235776311405, 10204120439288725, 99244762587719585, 990878067150790140, 10145281310155565842, 106420501631411705747, 1142671059786354295966, 12548652816798990883431, 140839029768184796119004

Offset: 0

Author

Paul D. Hanna, Mar 21 2018

Keywords

• nonn

Examples

			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 + ...
such that
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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A301465.

Programs

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)}
for(n=0,30,print1(a(n),", "))

PARI

{a(n) = sum(j=0,n, sum(k=0,n-j, binomial(n-j,k) * binomial((n-j)*k,j) ))}
for(n=0,30,print1(a(n),", "))

Formula

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

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1+x)^(n*k).

a(n) = Sum_{j=0..n} Sum_{k=0..n-j} binomial(n-j, k) * binomial((n-j)*k, j).