A217668 - OEIS

%I #21 Apr 25 2018 09:37:30

%S 1,1,2,1,3,1,5,1,5,4,6,1,14,1,8,11,13,1,25,1,22,22,12,1,61,6,14,37,50,

%T 1,77,1,73,56,18,36,175,1,20,79,211,1,135,1,188,232,24,1,421,8,236,

%U 137,313,1,307,331,422,172,30,1,1423,1,32,295,601,716,727,1

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

%H Paul D. Hanna, <a href="/A217668/b217668.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Sum_{n>=0} x^(n*(n+1)) / (1 - x^(n+1))^(n+1). - _Paul D. Hanna_, Sep 13 2014

%F a(n) = Sum_{d|n} binomial(n/d, d-1) for n>0 with a(0) = 1. - _Paul D. Hanna_, Apr 25 2018

%e G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 +...

%e where we have the following series identity:

%e A(x) = 1 + x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5  + x^6*(1+x^6)^6 + x^7*(1+x^7)^7 + x^8*(1+x^8)^8 + x^9*(1+x^9)^9 +...

%e A(x) = 1/(1-x) + x^2/(1-x^2)^2 + x^6/(1-x^3)^3 + x^12/(1-x^4)^4 + x^20/(1-x^5)^5 + x^30/(1-x^6)^6 + x^42/(1-x^7)^7 + x^56/(1-x^8)^8 +...

%t terms = 100; Sum[x^n*(1 + x^n)^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, May 16 2017 *)

%o (PARI) {a(n,t=1)=polcoeff(sum(m=0,n,x^m*(t+x^m +x*O(x^n))^m),n)}

%o for(n=0,100,print1(a(n),", "))

%o (PARI) {a(n,t=1)=local(A=1+x); A=sum(k=0, sqrtint(n+1), x^(k*(k+1))/(1 - t*x^(k+1) +x*O(x^n))^(k+1) ); polcoeff(A, n)}

%o for(n=0,100,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 13 2014

%o (PARI) {a(n) = if(n==0,1, sumdiv(n,d, binomial(n/d,d-1)) )}

%o for(n=0,100,print1(a(n),", ")) \\ _Paul D. Hanna_, Apr 25 2018

%Y Cf. A217667, A217669, A217670, A143862.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 10 2012