OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Sum_{n>=0} x^(n*(n+1)) / (1 - x^(n+1))^(n+1). - Paul D. Hanna, Sep 13 2014
a(n) = Sum_{d|n} binomial(n/d, d-1) for n>0 with a(0) = 1. - Paul D. Hanna, Apr 25 2018
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 +...
where we have the following series identity:
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 +...
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 +...
MATHEMATICA
terms = 100; Sum[x^n*(1 + x^n)^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)
PROG
(PARI) {a(n, t=1)=polcoeff(sum(m=0, n, x^m*(t+x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))
(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)}
for(n=0, 100, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 13 2014
(PARI) {a(n) = if(n==0, 1, sumdiv(n, d, binomial(n/d, d-1)) )}
for(n=0, 100, print1(a(n), ", ")) \\ Paul D. Hanna, Apr 25 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 10 2012
STATUS
approved